Diagnosis: Wiggle drive-side crank arm, then wiggle the non-drive side crank arm. If one side wiggles slowly or is loose, but the other side is firm, then it's a loose crank arm. If both sides have perceptible play of equal amount, more than likely, the BB is loose.

First get familiar with how a BB assembly basically works: They are all similar in mechanical design principles. There is a heavy-duty, usually round, beefy, horizontal steel spindle in the middle that spins on two sets of bearings on either end of the bottom bracket shell. To fix the bearings against lateral movement into the frame of the bike, the assembly includes bored cups that are threaded tightly (usually in the US, drive-side is left-hand thread, non-drive side is right hand thread) into the shell into which the bearings fit, which then sandwiches the spindle precisely inside the BB shell. The spindle is LONGER than the BB shell, and the ends protrude from bored center holes of the threaded cups. The ends of the spindle are either notched or square-tapered. Right and left crank arms mount onto the ends of the spindle, and are then pressed and fixed onto the spindle using crank bolts ( or sometimes nuts if the spindle has threaded bolt ends rather than a tapped threaded hole - like a belly button that's an insey, or an outsey)

If both crank arms wiggle: Usually this means BB cups are not tight up against the spindle. The cups will need to be adjusted and re-tightened to torque spec. Usually on the drive-side, I use between 30 - 50 ft-lbs. On the non-drive side, with lock-ring style (traditional) BB, I use about 20 - 25 ft lbs. With newer cartridge BBs, I use 30 - 40 ft-lbs. To tighten the cups correctly, often requires removing the crank arms off of the spindle. You will need a crank arm extractor tool to do this. Note that drive-side cups are left-hand thread (lefty tightey, righty-loosey). non-drive side cups are threaded normally (i.e. righty-tighty, lefty-loosey).

If one crank arm has play and moves in a viscous way, then it may be damaged already or too loose. If the crank is not damaged too badly, you may be able to tighten the crank bolts tighter and reach the torque spec. Otherwise, you will need to replace the crank arm with something compatible (i.e. same length and supports same BB spindle end standard - e.g. JIS square taper, ISIS, Octalink, etc.) If you tighten the bolt to spec, and the crank still moves and wobbles, replace that crank arm.

If you cannot tighten the crank bolts either because of being stripped or deformed, then you may need to replace the spindle (if it's threads inside are damaged) or you may need to get new bolts. Bolts are either 8mm or 10mm allen or 14mm hex socket. You will need to tighten these to 20 - 25 ft lbs of torque or other spec according to your maker to prevent slippage and the crank arm from loosening and possibly getting damaged. Over tightening can also cause damage so you need to follow specs.