My wedding I want to limit the alcohol intake because a few family members have drinking problems, and I do NOT want a bunch of drunks at my wedding. So I was going to limit one bottle of white and one bottle of red to each table of 8 people. Is that enough?

I think you should do 2 white and 2 red, because some people don't drink one or the other. If everyone drinks white, it will be gone with 4 people.

Or you could have the caterer replenish the empty bottle, but limit it to just one refill.

The standard rule of thumb used by caterers is half a bottle of wine per person (see this page on wedding drinks). If you are spreading this out over the whole meal, with food, then people shouldn't get too drunk. So I would agree with JD and go for 2+2 bottles per table. If you are using caterers then perhaps have someone point out the problem family members in advance, so they can go easy on the refils?

One bottle of wine is 4 glasses. I agreee that you should have two and two to allow for the taste of the guests. This will still limit it to an average of 2 drinks per person.