Possibly. It's hard to say, though. This has been kind of an extreme weather situation. We don't normally get smacked with so much rain in a week, for several days in a row, so we can't really say what normally happens because this is not exactly the norm. Certainly there will be more people in the parks when it is not raining and it will be back to business as usual, but unless there is some specific thing to draw large numbers of people next week (or when it dries up), it shouldn't be any more crowded than it would have been anyway on a non-rainy day in January, given all the new AP holders and what not. In other words, unlike the holiday season - where a rainy week can throw off plans to see the seasonal festivities within the limited time they are up - there is not really anything that new or out of the ordinary or only up for a short time that is happening at DLR this month, so I don't see why it would be abnormally crowded even when the rains subside.
This is an El Nino year, though, so we may not be seeing the last of the bad storms. Usually, when we get El Nino rain in California (as opposed to just regular rain, which is not much in a year), it rains a lot, for a long time, in California, and it goes into February, March and sometimes April, too. Since this bad rain is happening in January, I have a feeling we are not out of the woods yet!