Is there a better word than "helpee" to describe a person who receives help?

Not "the helped". There is almost no circumstance where that would be good English. Instead use "the one helped", "the helped one/person/etc".

Recipient and Beneficiary are totally different - they both mean receivers of something. In the case of Beneficiary, it is usually something good.
Recipient is neutral - one can be the recipient of a gold ingot, or the recipient of 50 lashes. Neither item is, strictly speaking, "helpful".

I agree with Martin Krzywinski's comment above to the effect that a person who benefits from your help can very reasonably and accurately be called the beneficiary. Merriam-Webster's Eleventh Collegiate Dictionary (2003) has this entry for the word:

beneficiary n (1662) 1 : one that benefits from something 2 a : the person designated to receive the income of a trust estate b : the person named (as in an insurance policy) to receive proceeds or benefits.

I'm not sure what negative connotations you associate with these definitions. I suppose you could argue that definitions 2(a) and 2(b) presuppose that someone must die or something bad must happen before the benefits begin to flow to the named beneficiary; that view isn't entirely true, but undoubtedly the connection exists in some instances. However, nothing of the sort applies to situation described in definition 1. For example, a person can be the beneficiary of someone's kindness without there being any need for anyone to suffer a loss.

The only other possible negative connotation of beneficiary I can think of is that it may seem too presumptuous a word for the helper to use. After all, "help" isn't always that helpful, and some attempts to help fail utterly. But when I receive real help from someone, I have no uneasiness about calling myself the beneficiary of that person's aid. And the relationship of helper to helped already implies the idea of a giver and a receiver of benefits, whether you use the word beneficiary or not.

Mental Health Case Managers are professional helpers. Referring to his/her clients as cases does not do the relationship justice. Manager implies staff; would it make sense to call the helpee, staff? There is a sense that the relationship is therapeutic. Therapeutic implies the helpee get the benefit of doubt (unless you have reason not to) and that the relationship is for the best interest of the client. Treating the client as making sense, is significant, acceptable, and has resources and strengths is essential.