And crossing the Atlantic Ocean made you prone to dying of scurvy. People still did it.
Besides, how do the astronauts on the ISS deal with radiation? Is the ISS somehow better protected near earth than in deep space?
yes but it's incredibly complicated, radiation comes from multiple sources. See https://en.wikipedia.org/wiki/South_Atlantic_Anomaly and https://en.wikipedia.org/wiki/Van_Allen_radiation_belt
Once away from low Earth orbit the shielding on a spacecraft becomes very expensive in terms of size and weight, even when it is the lighter weight polyethylene material.
> How are we going to deal with this?
I don't have an answer for the journey to Mars itself (and I'm also not sure if there's a consensus on how big a danger would such trip pose) - but you'll definitely want to keep the fuel between the passengers and the Sun. It would likely also be wise to pad the whole vehicle with some dense material - but to do that, you'd really want to have some sort of manufacturing capability in cislunar space, to avoid having to lift all that mass upwell.
As for bases on Moon and/or Mars, the best solution is probably to dig in - find a cave or make a tunnel, set up there, and develop underground. Ground is a very good radiation shield.
And probably treat cancers as they happen, accepting the risk of shorter lifespan.