My first choice would be to try to return it, or check into factory warranty period.
If you're just stuck with a meter that doesn't work and no options for returning it under warranty, your other options require you to have some troubleshooting/repair skill.
I hesitate to offer advice on this because if you have no experience with this, you're just going to get yourself into trouble and will make your meter become unrepairable junk.
Try to get it repaired under warranty if you can.
If you have no options, and want to attempt to find the problem, you'll have to open the meter housing to get at the electronics.
You won't be able to repair the electronic components if that's the problem. Parts will be surface mount and mostly unmarked.
The only reason I suggest this is so that you can check the mechanical contacts in case that's the problem.
Cheap DMM's have a really cheap spring metal contactor that moves with the front dial to select a meter function.
The push-buttons will be cheap carbon pads to contact the trace on the circuit board.
If you open the housing, use EXTREME CAUTION, as all the spring stuff and flexible plastics will fall out all over the place and you'll have a hard time fitting everything back together.
You said you were worried about moisture. What I'm suggesting is so you can check the ON/OFF POWER push-button contactor, and that sweep arm connected to the knob.
I'd use isopropyl alcohol and a cotton swap to make sure all the electrical contacts are clean, moisture and corrosion free.
You may not want to bother with this at all. If your POWER button contact is clean and your function selector contactor is clean; you probably have a component failure that you won't be able to fix.
Sorry, but that's what I'd do. Return it, get factory warranty to fix it, and only open the housing if you have no choice but to discard it because it doesn't work, and want to open it up as a last resort.