On what basis do some denominations prohibit Christians from drinking alcohol? From my understanding the bible is quite clear that Christians should not get drunk. However some denominations seem to go a step further and say that Christians shouldn't drink at all. From what I can work out the Old Testament Law doesn't prohibit drinking, and I can't find a record in the NT banning it.

How do any of these denominations support their belief that Christians shouldn't drink at all?
My denomination is one that is very strongly in the "Christians shouldn't drink - at all" camp.  Very strongly.  And my Pastor, whom I love and respect, is very strongly in that camp as well.  Yet even he, in his sermons, will tell you that nowhere in the Bible is drinking explicitly called a sin.

However, in many places, drunkenness and even alcohol consumption is called foolish.  And to him, and every other Baptist Pastor I've heard preach on the subject (and that's quite a few), the problem isn't so much that drinking alcohol is a sin, it's that drinking alcohol lowers our inhibitions, and sinful behavior often follows.  It's really easy to give into temptation if you're under the influence of alcohol.

That said, there are other theories about the prohibition against drinking.

From http://www.brucesabin.com/alcohol.html


  In the history of Christianity, alcoholic prohibition is a relatively
  new idea. In fact, alcohol was a normal part of life. In Colonial
  America, the Puritans expected Christians to drink (Hearn, 1943). In
  the 1700s, a Baptist minister created the formula for bourbon whiskey
  (Hailey, 1992). During the 1800s, many Southern ministers operated
  stills, and sold alcohol (Hearn, 1943). Parishioners who owned stills
  would tithe their alcohol; and preachers' salaries often included
  whiskey. All this began to change, however, as the Temperance movement
  took shape (Hailey, 1992).


So even in Baptist Churches, it hasn't always been that big a deal.

However, the article does go on to echo some of the reasons for abstaining altogether.

One that I've heard from the pulpit, and that I personally can't argue with is echoed here:


  For some Christians, a sense of love and justice leads them to abstain
  from alcohol. Seeing alcohol's devastating effects on society, these
  Christians feel compelled to act. Taking a strong stand against
  alcohol and its consequences is a tangible way that they can
  demonstrate their faith. These Christians believe that abstaining from
  alcohol is the best, possibly the only, solution to this societal
  problem. In addition, abstaining from alcohol guarantees that one will
  never fall prey to alcoholism (Hancock, 1999).


Quite frankly, even though most people never have a problem with alcohol, it can ruin lives.  And in the spirit of 1 Corinthians 8:9 (King James Bible (Cambridge Ed.))


  But take heed lest by any means this liberty of yours become a
  stumblingblock to them that are weak


It's just considered a good idea to set a good example. As my Pastor often says: "People are always watching you.  The lost are always looking for an excuse to say, 'He's not a real Christian - I see him down at the bar all the time  Who is he to try to act all holier-than-thou?'".  In the interest of maintaining a good witness, and staying credible, it's simply more constructive to abstain.

As an added list of reasons some Christians believe that it's wrong, I'll include this link: http://www.scionofzion.com/drinking.htm

I am not saying I agree or disagree with any point on the list.  I'm also not debating what my Church preaches is Truth. I'm just presenting it, like the rest of this answer to address why some believe drinking is wrong, not addressing whether drinking is wrong. My own views aside, this isn't the place to determine such things.