Rain, wind, and even thunderstorms are pretty common in DC in the spring. So this is a question that comes up often during cherry blossom season. Here's my general response and the reasoning behind it.
Heavy rain and strong winds can knock petals off, so there is factual basis for concern. That's especially true a few days after the peak bloom, as the flowers become even more fragile. Here's a vivid example from heavy rains a few days after peak bloom in 2015. And here's a very similar scene from 2014. In both cases, it was about 4 days after peak bloom.
When the trees are approaching or even in full bloom, they're less fragile (to rain, at least--frost can be a different matter). And even when in full bloom, there are so many flowers to start with that it take a lot to be knocked off before you even really notice. So while it's not impossible for a severe storm to make a big dent in them, the effect is usually much less drastic.
Of course, I have no way to magically predict the weather, and severe and unusual weather is in fact possible. If we happened to get something like a derecho sweeping through the area at that time, all bets are off. But in general my answer to this question when it comes up is that I wouldn't let rain or storms in the lead-up and even during much of the bloom dissuade you from visiting. Chances are very good that there'll still be plenty to see. And there's always a reasonable chance that the weather forecast is wrong, that any storms in the area prove to be very localized, or the rain isn't as heavy as expected. Yes, there might be petals on the ground. Maybe even a lot of them. But that can also be pretty in it's own way.
However, if the rain, storms, or wind are coming 3 or more days after peak bloom, it's a different matter. Then it can have a real effect and in the right combination of conditions the branches can be stripped nearly bare. But again, the risk of that is mostly several days after the peak bloom.
For more examples, take a look at the What to Expect, When post.
Want to Help Support DC's Cherry Trees?
If you'd like to help support the care and upkeep of the cherry trees around the Tidal Basin, the Trust for The National Mall has launched an Endow a Cherry Tree Campaign. Donations go to the official Cherry Tree Endowment, which will give the National Park Service additional resources to fund the care, maintenance, and possible replacement of the cherry trees. You can find more information here.
The Trust is dedicated to marshaling private support for maintaining and improving the history National Mall area. I'm not affiliated with the Trust--just an admirer of their efforts.