Be aware of them beginning with the loading screen right up until the point they're dead. Make note of when, where and especially if they were spotted before you commit yourself. Don't spend the whole game fearing a DM will be around every corner and play passively as a result, but don't play recklessly when that DM hasn't been spotted for the last 5 minutes.
I personally always go for the cap at the start in most DDs, but will turn around and sit still right on the edge of the cap circle facing back towards my spawn in case I need to make a speedy exit. That can be caused by either a radar or a stealthier DD. Sitting behind anything but the most substantial cover or in smoke is a big no-no because it will deny you vision of an incoming radar ship but the fact that the cap is progressing will tell the radar ship you're in the cap, plus with smoke you're just asking for a torpedo salvo to head your way. I'll always contest the cap at the start, but I'm ready to leave it at a moment's notice and come back to it later if necessary. That's not camping, that's simply not being suicidal.
There are also common sense ways of avoiding being radared. A radar has a short duration and a long cooldown, radar ships won't use in randoms it unless they're fairly confident of spotting something. Even if you're inside an enemy ship's radar range they won't necessarily know that unless you do something such as cap, shoot or smoke to give your position away. You don't even necessarily need to get out of radar range immediately, just don't loiter well inside of a potential radar bubble with no exit strategy or plan to survive the duration. There are a few ships in the game who are basically able to radar the moment they're spotted because their detection range is basically identical to their detection range, those are ships you simply need to be aware of and avoid putting yourself into positions where a single radar duration from them could kill you. At the end of the day radar does nothing but reveal your position. If all it took was for you to be revealed for you to be killed within seconds from a high level of health, you were probably horribly out of position and running into an enemy DD would likely have had the exact same result.
Everyone dies to radar, everyone fails to pay attention or stumbles into a radar ship they didn't think would be there. There's nothing inherently imbalanced about the mechanic though, even if we're approaching a slippery slope in terms of numbers per game and distribution across teams. Mistakes simply hurt, learn from them and strive to make less.