In Denali Park a couple of years ago we got the crash course in river crossings.
Some of the earliest replies have already mentioned some important points: use poles, cross in groups (a line with one person breaking the current and everyone else in his "shadow" is best) and support each other by linking arms or grasping shoulders, turn sideways to the current, wear shoes. Proceed very slowly, taking small steps and always making certain of your footing (and your poles). Don't look at the water rushing by; look at the shore you're aiming for, or the person next to you. Don't fight the current, try to proceed across but plan on drifting downstream. (Leave room for this, don't cross just upstream of rapids.)
Before stepping into the water, unbuckle the waistband of your backpack and loosen the shoulder straps - if you do lose your footing, a heavy pack can drown you quickly even in shallow water. If you do fall, remember the earlier advice about riding in "safe" position versus trying to stand up too quickly.
A further skill is choosing the best spot to cross. You don't want the narrowest spot, you want the shallowest and slowest, which is usually also the widest. You can tell deep spots, strong currents, and big rocks by their effects on the surface ripples - it's hard to describe but not difficult. You can also predict current based on the shape of the river and its bank. For example, a sharp bend will be almost always accompanied by a strong and deep current along the outside bank, and that bank will be steep or even overhung.
If the water is cold, put on a jacket and a hat before you step into the water. Once you're in the stream, forget about the cold and just concentrate on getting safely across.
PS as you proceed, test the depth with your hiking pole. Depending on current strength and slipperiness of bottom, waist-deep is probably the upper limit for safety. Anything deeper and you had better be prepared to swim (NOT recommended in the presence of current and rocks)! (See picture below: barely knee-deep and we're having some difficulty.)
PPS Using poles is much more difficult than you might think, because the current pulls at the pole too. But they're tremendously useful for keeping your balance.