Lego Arms/Wrists getting loose?

Hmm... Maybe the hand/wrist joints are getting worn out from frequent swapping? Perhaps new hands is the way to go. Are you sure that the arms aren't cracked at the wrist?

Honestly, the best solution is to just get new minifig hands, or arms. (To check which is the problem, just try two or three pairs of hands and if all of them are loose, the problem is the arms, otherwise you likely just have a bad pair of hands.)

This just happens occasionally, especially if you pose the hands or change hands excessively. What it is is the friction between the pegs on the hands and the sockets wears away the plastic over time and thus- less friction. It occurs more often than it used to because there is increased production of minifig parts in china, which all have less durability than their European counterparts. The same thing happens with legs, and ocassionally arms (torso sockets). I never use any Chinese made hips for precisely this reason- they simply have no grip.

To fix what you already have, the best solution is to insert a very small amount of superglue inside the arm socket and then stick a hand in, turn it back and forth for 5 seconds and pull it out. Wait for it to dry and you'll have some friction back. Sadly, while this is the best solution, that doesn't mean its a good solution. Its hard to pull off correctly, and even if you do, its only temporary, as continued movement will cause the glue to crumble and you'll have to do it all over again.

while the suggestions above will work, they are honestly not the best one out there...

use clear nail polish,

for loose heads, legs, hands, arms ect, a coat or two of clear nail polish adds a smidge of thickness that allows a better bond to the worn or mis-cast pieces. paint it on the male end of the connection (neck joint peg, hand peg, ect) and let it dry, 5-10 min is normally well enough, i do two min of either blowing on it or holding it near a lamp.

works great, cheap as all hell, and does not destroy or permanently seal the piece.
edit: i forgot to mention that this is more effective than the superglue method as clear nail polish is normally made of a acrylic which never get crazy solid or crazy brittle, like the super glue does, making it less likely to flake off.