Ordinary physics obviously doesn't hold much sway over these people. I have to assume some "brings his own dimension" shenanigans like the Flash does to give the speedster the time to walk, turn, open doors etc without instantly breaking themselves or the thing they operate.
Solution 1: a big heavy door and reinforced walls and an airlock.
They might be fast, but if it would take days or weeks to open a door without the key or actual vehicles designed to wreck the place, they will take long enough that you can deploy countermeasures against them. Let's say you increase the pressure inside until the speedster is knocked out. That way you circumvent the speedster having prepared with gas masks or similar. Without gas masks the fact that they entered an airlock would mean they use the air up so quickly that they'll knock themselves out before they breach the door.
Solution 2: electrify it.
The problem with being a speedster is that all your internal processes (and a few external one's) have to be accelerated as well. This means that electricity will pass through you that more quickly while dealing the same damage as before. A sufficient jolt of electricity can knock the person unconscious, or have him writhing across the floor as the speedster finds out that his rubber boots protected him from the floor until he fell down after grabbing the door handle. If you are less friendly, just kill them with a high enough jolt.
Solution 3: low resistance surface.
If the floor is so slippery that a normal person cant get up anymore, for example because the floor is bowl-shaped and outlets make sure a nice film of ultra-slippery material coats it constantly (or it is slippery in normal circumstances, like a thick layer of ice you've cultivated), then a speedster wouldn't be able to get up either despite his speed. If you are friendly you have softer materials at the edges to make sure the speedster doesn't kill himself with the impact before sliding back to the center and getting stuck.
Solution 4: mines and boobytraps.
The speedster has to accelerate a part of his environment in order to move that fast. Without that the friction between the floor and his shoes would limit his speed to around 50km/h (I did a question about this once), and that is ignoring the fact that his shoe rubber would likely split apart. So his acceleration has to extend to his shoes and the ground he's standing on as well in order to reach his speed... and if he steps on a mine trigger, that will be accelerated and go off as well (not that it needs to! The chemical reaction upon standing on the trigger will be fast enough!). Mine shrapnel like from a claymore travels at 1200m/s, faster than 2000mph (+/-900m/s). 300m/s difference isn't going to give even a speedster much chance to react. Similarly, a door with a bomb strapped to it will kill the person if he opens it.
Solution 5: be somewhere their speed is irrelevant.
Let's say that the moment the evil speedster starts trying to kill you, you are in an aircraft crossing large bodies of water. How is he going to catch you? If he swims he'll die from starvation before reaching the other side, boats are too slow and unless the speedster knows how to operate functioning military aircraft he's not going to be able to catch you in the air. And that is ignoring the fact that he might be able to get to an operational, loaded and fueled aircraft but said aircraft would still need to pilot to the runway and take off, then avoid interception at aircraft speeds rather than your own until he reaches you.
And that is assuming he even knows where you are.