If we judge from the position of realism, then it's more likely that you'll catch a virus from a tree rather than an alien - your DNA has more in common with a banana than any alien out there.
So things that left are microbes and microorganisms. Even if they won't eat your body directly, they can snag up nutrients, or mess up your functions with their excretions (Imagine if you had contracted microbe that eats sugars and poops out arsenic)
What's the possible solution to that?
Aliens have artificial immune and microbiota systems.
In my story aliens were faced with the problem of microbiota cross-contamination while dealing with each other. Even if bacteria have no ill effect on the crew, it can devastate a biosphere it's introduced to if it begins to outcompete natives and starving them out by gobbling up all the microelements they need, while itself being incompatible with the local biosphere for it to make use of the new invasive species. This will cut off the food chain at the very root, which will destabilize and destroy the entire ecosystem in very short order (I stumbled upon that problem being highlighted in the Rifters trilogy by Peter Watts, by the way).
The ships themselves are routinely sterilized by a variety of methods to ensure no bacteria gets carried from a planet to a planet, but this still leaves another vector of contamination: the crew itself. Even if you bathe in alcohol disinfectant every hour, there's still about a quarter of a kilogram of bacteria living inside you. The problem? You can't just kill them off, they're symbiotic and vital for our functioning, mainly the gut microflora.
So what to do? My aliens went with the augmentation route. They have a large variety of implants that take on the role of the body's natural immune, digestive, and a few other systems without compromising health, basically filling themselves up with nanotechnology - the implants themselves aren't nanotechnological, but they can manufacture and distribute nanoagents and microscopic robots for various purposes. This results in the aliens being basically sterile-clean inside and out at any time (and the poop doesn't smell too). Any bacteria that gets in is almost immediately targeted and destroyed by a computer-controlled artificial immune system, and any bits of these systems that end up outside of the body become inert pieces of indigestible material with zero impact on the local ecology, as they lose both the control signal and energy supply of the host. what's great about this system is that it also improves the overall health of the individuum as a side-effect (getting sick is virtually impossible now, and even some health conditions can be targeted and fixed before they'll become an issue).