No, the solution is NOT in preventing the water from coming up.
The solution is in clearing the drain pipes so the water in your drain piping can flow away as quickly as the washer pumps it in. What's happening now is that the washer is pumping it in faster than it can drain away through partially clogged pipes, and that's why it's backing up into sinks and such.
Go into your basement and stand under where the toilet in your house is. You should see a big pipe within a few feet of you that goes straight down into the concrete floor. On the concrete floor near the base of that pipe, you should see a clean out. You need to run a snake in there and clear the main drain line from your house first. Ideally, it's best to clear it all the way out to the city sewer under the middle of the street, but most of the blockage will be within 20 feet or so of the base of that vertical pipe. Clearing the gradually downward sloping main drain line that comes off the bottom of that big vertical pipe should alleviate much of the problem. But if some problems still persist, you might have to clear individual drain lines in your house.
What happens is that there's so much water pumped out by the washer so fast that it tests the capacity of your drain piping and becomes the "canary in the coal mine" to let you know if the drainage piping is starting to get clogged. Your washing machine is trying to tell you to have the main drain line from your house cleared, and it's a good idea to do this every 10 years or so just as preventative maintenance to avoid drainage problems in future.