Having played all three (alto and basset at some length, and basset clarinet for a couple of hours at an ICS convention), I can understand why the alto and basset are not day to day items. I didn't make up the comments "vapid" and "insipid", both of which have been applied to the alto horn; people with far more musical qualifications than me have dissed same. I also happen to agree, mind you...
The basset horn is a bit more problematic. Here, the problem may be more a matter of a lack of skill in players of same, as much as it is due to the design and capabilities of the horn. There are good basset horn players (I was once pretty damn good on a Selmer one; alas, that is no longer the case), but they are thin on the ground. Many universities have basset horns in the horn room, and when the odd part comes along, they draft some poor schlub like me to cover it and then let it go until the next time. Most of us are having enough trouble getting a quality bass clarinet in our hands, much less a horn that gets used once in a blue moon.
In any event, the demand for basset horns is largely governed by the number of parts available for them. Modern composers aside, there are precious few basset horn parts out there (and almost all of them are in small ensembles). Even throwing the moderns into the mix, it's just about as bad. (So too are the number of alto clarinet parts outside of concert bands and clarinet choirs.) Just as I don't own an Ab clarinet, I don't see much need for my own basset horn. It doesn't stop anyone else from buying one, but damn'd if most others don't buy them as well...
I am also puzzled by the attention given to the basset clarinet. In some part, it's interesting because it's there, just like the Ab horn. Novelty has a fascination all its own for some. And, there is some legitimate call for a transistorized bass clarinet (for that's what you have in a basset clarinet), since there are some strong indications that Mueller played one back in the day. Modern mechanism made them possible (whereas Mueller's may have only barely worked, and thus died on the vine), and the Sax designed bass clarinet extended way down low served as a prototype.
However, just because something can be done doesn't necessarily mean that it should be done. Many trombone players view the valve trombone as one of the greatest abominations onto God that has ever been perpetuated, and "octo-contra-bass" clarinets probably fall into that category as well. Some clarinet players feel this way about the saxophone, for that matter...
(And, where are the voices calling for the clarinet in G? The clarinet in D, so that Strauss can be done "authentically"? Why doesn't anyone rise up when the contra-bassoon is used to play the ophecleide part in The Sorcerer's Apprentice?)
There's more call for it now, since some modern folks have written works for same. However, most clarinet players are more likely to encounter a bass clarinet in C part than they are a A clarinet part to low C (outside of the "revised Mozart", of course). Does a .0001% chance of running across one of these parts merit the expenditure needed to carry a basset clarinet (not horn) home? Sales figures answer this one for me, just as they do for the bass clarinet in A. You can buy them, but precious few are doing so...
Mind you, I feel much the same way about the bass clarinet in Bb to low C. When I bought mine, I had the bucks laying around (enough to buy it and a sailboat at the same time, thank you); had money be tighter, I probably would have gone with the short horn.
And, I don't begrudge those who want to play basset horn, any more than I do those who want to play alto clarinet. There is a legit literature for basset horn, and if you've got the money (or have someone to bankroll you the cost), more power to you. (Alto clarinet has the added advantage of being available for next to nothing for a second-hand "pro" instrument; hard to argue with that economic factor.)
But, like other "relics" (the serpent, to point one out), these horns have (by and large) been passed by in the grand scheme of musical things. The oboe 'd amore is a perfectly good instrument, but it (along with many other instruments, like the C melody saxophone) has been marginalized.
Some of this is tone color (personally, I cringe whenever I hear the alto clarinet played, even when I'm doing it myself), some of this is economics (have you oiled your Ab and D clarinets yet this year?), and some of it is the milieu in which they operate (where, oh where, are all of those bass clarinet parts to low C?), but it ends up being (in practical terms) an accomplished fact, like it or not.