What can I do besides...

The word "beside" is a preposition, therefore it has no plural.

To my ear, the sentence "What can I do but complain?" sounds best. It is concise. It avoids the plural preposition and the unneeded gerund.

@dogreed
Sorry dude, you got it wrong.
Besides appears in both the OED and Chambers:-
besides prep
in addition to, as well as or apart from something or someone.
adverb 1 also; as well • We saw three more besides.
2 (often as a sentence connector) moreover; in any case • I don't want to go;
besides, I'm not dressed.
ETYMOLOGY: 13c: from, and originally with the same meaning as, beside.
The OED entry is somewhat longer.

@Anwulf
I agree.
"What can I do but complain?" does sound better, as does "Besides complaining, what can I do?"
In fact, at the risk of incurring your wrath, both those forms sound more elegant.
Perhaps location is the key?
"Everyone, besides John, wore green." vs "Every one wore green besides John"

I think we all agree that the first one sounds wrong. This was raised by an ESL learner. I told her that the second two were better. I was just trying to put my finger on why.

@Dogreed, I think the "but" works better here as well.
@HairyScot ... I agree. Word order matters in English. In the end, I came to the same thoughts.

Thanks!

To answer the original question of why: What can I do besides complaining sounds wrong because can and complaining conflict in their tense. You can complain. You can't complaining. You can *be* complaining. (Although it occurs to me I haven't got it quite right... 'do' could be the operative word... but you can't 'do complaining' either). Would what can I be besides complaining work?

Either way, complain is an action, complaining is a state of being. That is why your in-ear English radar picked up on it. Both can and do imply action. What can I do besides living? That just doesn't work. What can I do besides live? Much better.

(Note that you can replace all instances of 'besides' with 'but' and I'm fairly certain that all that I have said will still stand true)