The best configuration for the low note keywork on basset horns, basset clarinets and low C bass clarinets is a subject of much discussion, and different makers offer various and often very different solutions.  While there is no true right answer to this question, below are some thoughts which may be of value in considering options.

Two fundamentally different concepts exist:  multiple keys for the right thumb, and avoidance of thumb keys in favour of additional little finger keys.

Early basset horns, from the origin of the instrument in the 1770s until its temporary hibernation starting in the mid-19th century, without exception had all basset notes accessed by keys for the right thumb.  The number and arrangement of the keys was extremely varied - as was the matter of which pads were normally open and which closed - but thumb keys were universal.  The same was true of "basset clarinets", which during this period were simply basset horns pitched in A or higher.

Basset horns of the modern type from German makers, from the late 19th century up to the present day, likewise as a rule have either four right thumb keys for the basset notes, or three thumb keys with low Eb for the right hand little finger.  German low C bass clarinets are made similarly.

When the first modern basset clarinets were made - first in Czechoslovakia in the 1950s and subsequently in Britain around 1970 - the German basset horn pattern was followed.

In contrast, the preference for little finger keys instead of thumb keys has been a constant feature of basset horns from French manufacturers throughout the 20th century, and of basset clarinets since these makers started offering them circa 1980.  There is no uniformity of specific key locations and assignments (other than the normal use of a thumb key for bottom C), and no attempt will be made here to catalogue the various arrangements used.  Not only does each maker do things differently, but even the same manufacturer will vary the pattern among different models of instrument.  (One may or may not choose to criticise the lack of logic in this approach!)

Prior to the mid-1980's, low C bass clarinets from French makers normally had the Eb key for the right hand little finger, on the lower rank beside the RH E/B key - the location sanctioned by many decades of tradition on instruments to low Eb - and three thumb keys for D, C# and C (usually not constructed ergonomically).  More recent models show the same multiplicity of key arrangements as on basset horns and basset clarinets.

Three sets of criteria are used to determine the best way of handling the question:  (1) what the music requires; (2) what the player can most efficiently learn and operate; and (3) what is simplest, most reliable and most practical for the instrument maker to provide.

A full discussion of what works best for various musical passages would be endless, but it can be largely circumvented by one overriding observation:  the music for basset horn and basset clarinet dating from the Classical period and the early 19th century, and subsequently German basset horn music from around 1900, was all written to be played on instruments with four thumb basset keys.  Since this comprises virtually all the music of interest for most players, the conclusion is that some arrangement of thumb keys would be at least a safe and sufficient choice.  It is true that there are a few musical examples (in particular those involving trills) whose negotiation may be difficult to comprehend, but they absolutely must have been playable by professional musicians on the instruments of the time and place in which the music was composed (though there is one Db-Eb trill in Richard Strauss' Daphne which is questionable; see below).  Simply put, four thumb keys will allow any conceivable note-to-note transition, though perhaps not with optimum facility.

When one starts to replace thumb keys with little finger keys, some passages become easier, but others become more difficult or even impossible.  Note that not only must bottom C, C#, D and Eb be considered, but also E, F, F# and Ab.  Since duplicating all the basset notes for both little fingers is a nonsensical idea in reality (that would mean eight keys for the right hand little finger and nine keys for the left!), one is faced with deciding which note sequences will be encountered in the repertoire, and which one can assume will not arise.

It is regrettable and rather incomprehensible that the basset keys on some instruments seem to have been designed by technicians who had no experience with or knowledge of the standard repertoire.  Perhaps the most glaring example of this is the passage in the Mendelssohn Konzertstück in F minor involving a smooth transition from bottom Eb to Ab; this is impossible on some basset horns, for example the Selmer.

The arguments for opting for little finger keys seem to be (1) lack of confidence in the ability of the thumb to slide between keys, and (2) reluctance to release the thumb from its function of holding up the instrument.  Both of these arguments can be countered easily:

With properly designed keys (a significant caveat), and with a bit of practice, the thumb is perfectly capable of sliding securely and precisely between adjacent keys.  (If there is any doubt about this, simply consider the demands that clarinet technique already makes on thumb of the left hand, or the multiplicity of keys that each thumb of a bassoonist is required to operate.)

For a basset horn or bass clarinet supported on a floor peg and played while seated, the right thumb is clearly free to operate keys.  For a basset horn or basset clarinet played standing, releasing the thumb from support duties requires a support mechanism of some kind - a neckstrap or chest harness, preferably in conjuction with a belly brace or equivalent.  Such supports are available, however, and would be strongly advised in any case to avoid hand and wrist strain when playing such a large, heavy instrument.  Thankfully, it appears that prejudice against using support devices for clarinets, based either on the belief that they restrict one's freedom or just on macho reluctance to admit weakness, is finally declining.

A further argument against little finger basset keys is that for players with smaller hands, some of the keys are simply impossible to reach comfortably.  (This is especially a problem with some bass clarinets.)

Mechanical considerations with respect to basset keys are varied and sometimes rather subtle, and regrettably not everything the player might desire can necessarily be constructed satisfactorily.  Flexibility and weight of metal parts, moment arm distances, spring tensions, friction and lost motion in multi-part mechanisms are all concerns.  The instrument maker generally prefers simplicity and directness.  Smooth, postitive, light action and excellent reliability are goals of both the maker and the player.

The present writer admits to a preference for having the low Eb key in the traditional place (for the right hand little finger) and three well-designed thumb keys for D, C# and C, with the recommended addition of a left hand Ab/Eb lever.  Optional additions are a left hand little finger low D key, and an extra key for the right hand little finger which can produce either low D or C#, or be convertible for either function.  This arrangement will handle any musical passage one is likely to encounter, and is backed up by more than ten years' experience dealing with basset horns and basset clarinets as both a player and an instrument maker.

It should be noted that Strauss, though a master of orchestration, sometimes made mistakes when writing for unfamiliar instruments.  For example, the heckelphone part in his Alpine Symphony descends to F, four notes lower than the range of any heckelphone ever built!  Awkward passages in his basset horn parts may be put down to such lack of attention to detail.