If you are using a large 8'x10' tarp, it usually isn't much of a problem, though the large size can be a disadvantage in strong winds. My current solo sized tarp does need a bit more care in rain since its shorter in length. My worry is more about the head end then the foot end since my tarp is much shorter at the foot end since its not a true rectangle as you don't need headroom on that end. Due to being a cat cut tarp, I only use an A-Frame type setup though I may have the wind side staked directly the the ground while the other side may be staked high up in the air.
If in wind, you normally stake the tarp lower to the ground (lower your trekking pole height or tie it lower to a tree and make the tarp wider but lower to the ground) so that it acts less like a sail. This also means you'll have less rain blown under since the opening at the ends is shorter. In really strong winds, this may mean you are crawling into/out of the tarp. As Buck stated, you normally look for a sheltered spot, if available where boulders, trees, brush, logs, will block off some of the wind. Staying dry in a tarp is mostly about campsite selection.
I normally use my treking poles at the ends and stake it to the ground, but if rain is threatening and a large tree is available, I'll often tie the head end to the large tree instead which blocks a large part of the opening. In the past, I've put my packcover on my pack and used it and my rain coat to block off some of the opening after the wind shifted. I know some hikers who like to carry umbrellas for sun and rain and they use those to help close off part of the end.
I've been tarp camping since 2006. I've only been bothered by wind blown rain inside the tarp a couple of times when I was forced to camp in less ideal campsites with strong winds. Worse case, you collapse it and rotate the tarp around and put it back up. I normally don't bother and only did so maybe twice since the wind direction was often changing. Only 1 time did I get really wet underneath my tarp and that was due to a really poor campsite selection that I will never repeat. I normally carry a water resistant bivy sack for such situations and because I normally cowboy camp if it isn't raining.
On the plus side, a tarp has less condensation issues than a tent which can keep you drier in some conditions. When I hiked the PCT through Washington in 2009, I seemed to stay drier with my tarp then many other thru-hikers in their single wall tents and tarp tents due to this issue.