Why exactly would someone in the 13th or 14th centuries be given a nickname from a type of fish? Yet this is one of the possible explanations provided for the origin of Basset, from bass. Possibly more credible is a derivation from Middle English or Old French bass, meaning low, short or humble.
The name Beard looks more obvious, although at a time when razors were even less efficient than nowadays, it may have been reserved for someone with a particularly full or large beard. Like Basset, the surname was in the 1500s present in several districts across Cornwall, a sign of multiple points of origin.
More widespread was the surname Best. The origin of this is not what it might first appear to be to a modern observer. Unfortunately (for people with this name) it did not arise from the superlative of good but the Middle English (and Cornish) word for an animal or beast. The good news is that the nickname was not necessarily applied to someone who resembled a beast but who looked after them, i.e. a herdsman. (Compare the surname Heard.)
Cornish studies resources 
I found this in relation to Basset hounds: The name basset is derived from the French word bas, meaning ‘low’, with the attenuating suffix -et—together meaning ‘rather low’.
I found Best to mean ‘cow herd’ (as you say) but apparently it is possible that it could be applied (in its meaning as a beast, in a jocular, accepted way it seems) to a tough or brutal person.
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