Space

My understanding of the lore is that Martians got so good at fighting pathogens, they eradicated all of them, which lead to their immune systems gradually weakening over generations.

To be honest: I don't think that's much of a problem (unless...). The pathogens on earth are all adapted to attack (if you want to use this verb on a cellular level) us and others on earth. They would most likely not pose a threat to any other lifeforms that have evolved on other planets, unless our way of evolution is the only one which is able to produce life. And that is a big unless, because apart from the panspermia theory (life came to us with a meteor) there is no reason to assume that life has to work the same way it does for us.

It's the pivotal moment in War of the Worlds, for instance, and I can't see any way to avoid it.

I can think of a way to avoid it. And it's obviously a better first contact scenario when you think about it. It's simple, when you arrive at a planet for first contact, don't make it on the planet, make it at the largest space station found in orbit.

You can literally make contact in space suits, in space. If initial contact goes well and their required atmospheres (and necessary atmospheric pressure) are compatible, you can make further contact in the station, limiting the spread of any possible contamination.

Additional benefit: first contact will be made with scientists and engineers, not with politicians or nut jobs.