Get out of combat
Of course. It's a basic requirement for drinking, but it's easier said than done, specially when you have two Rogues slicing and stabbing your mana-starved butt. This is probably the hardest part, specially for classes with no natural ways to escape their opponents. Sometimes you'll need your teammates to peel attackers off you just so you can get out of combat. It takes five full seconds of you not attacking or being attacked to get out of combat, so it's a fairly small window. Don't cast anything on your teammates such as buffs or heals, as this will refresh your in-combat timer. Once you manage to shake off your pursuers, even for that small period, you then have to find the right spot to drink in (relative) peace.
Get some distance, get out of sight
You have to know the maps and be aware of where your opponents are. You should use abilities like Ghost Wolf or Blink, or even the Travel Form so ubiquitous in Arenas to put as much distance between you and any attackers. With any luck, they'll swap to another target or be crowd controlled. Once you've pulled away, find a spot safe enough to drink. In some cases, these are behind pillars or under bridges, but sometimes, specially with Shadowmeld, you can get away with a sip or two right out in the open. It depends on the situation you're in, but getting out of line of sight is always best. The pillars in Nagrand Arena are natural escapes, as are the four pillars under the bridge in Blade's Edge Arena (and the wooden pillars beside the ramps), and the starting areas in Lordaeron are common places to sit for a drink.
Don't lose focus and don't be greedy
Even when you've sat down for a drink, don't forget that you're still in the midst of a battle. If you're out of combat, your teammates are likely in the thick of it and could use your help. Particularly for healers, taking some distance and breaking line of sight could also mean that you won't be able to get back to your teammates in time. This is one of the most critical points of drinking -- knowing when to drink. It's a judgement call. As a healer, if your teammates are getting beat upon and you're running out of mana, what do you do? Conventional wisdom dictates that you heal until you're at 0 mana, hopefully topping your teammates off to give them enough to last while you drink.
Unfortunately, when healers (or other casters, for that matter) are out of mana, opponents usually call for a target switch, making it doubly hard to escape. The most sensible thing would be to heal until you're just a few spells short of bone dry. You'll need five seconds to get out of combat, anyway, so the last spell you cast should ideally not leave you absolutely drained. Don't wait until you're running dangerously low before you drink, and conversely... don't drink too long. You will almost never get the benefits of a full duration drink (although we saw some massive drinking sprees during the MLG tournament), and sometimes those four precious seconds -- the minimum time to get the effects of a drink in Arenas post Patch 2.4 -- are enough. Ideally, you should get as many ticks in as possible without letting your teammates hang out to dry.
Lastly, don't forget to bring drinks. The worst thing that could happen during a match is finding yourself free and out of combat but without anything to drink. Unless you're a Mage or teamed with one, Star's Tears are the only drink (for Level 70s) allowed in the Arenas. They can be purchased from Brave Stonehide in the Hall of Legends or Master Sergeant Biggins from the Champions' Hall for 25 Silver and 2 Honor points for five. These aren't soulbound, so you can trade them to teammates (sometimes even during a match!). Make sure to have an adequate stock, as the longer matches can see you drinking a lot.