Checking with your hyrdometer to see that you've hit your FG (and it has stabilized) is really the best way to tell if fermentation is complete.
The other method is waiting until a 'safe' amount of time has elapsed, so you can be relatively sure the beer is done fermenting. A standard rule of thumb is 3 weeks (1 in primary, 2 in 2ndary), but different beers may require more or less time. But then, this is only a rule of thumb. The hydrometer is science.
Yes, opening the fermenter does introduce the potential of exposure to infection, but if you are using the Hydrometer testing method, it is the only way.
The little bit you disturb your beer while checking won't have any impact on it in terms of oxygenation. Be delicate and don't get crazy with splashing in there. You can use your sanitized turkey baster. I use a wine thief, but it is essentially the same thing.
Just make sure to adjust your hydrometer reading for the temp of the beer. Most hydrometers are calibrated at 60F, so if you test at 70F, you'll need to do a quick adjustment. There are a number of online temp adjustment tools online, as well as in book form - like in How To Brew by Palmer.