Honestly, I don't see this as being a significant problem, as already alluded to by other answers. There are a few reasons for this.
1) Diseases evolve to attack particular hosts, and as such can not easily spread to other hosts. Viruses are not going to mutate to effect non-human lifeforms; which won't even have DNA to 'take over'. Bacteria and parasites that are harmful to humans are ones that evolved to live within a mammalian body as well. They are not as tied to their host, but their still evolved to a certain enviroment and it's unlikely any alien species will have a body that is a good hosts. Keep in mind that odds are alien life forms won't even be carbon based, they won't have the right nutrients for parasites to 'eat' and will likely have internal body temperature, acidity, and materials so foreign that bacteria can't live within it. In short most diseases can not spread from human to non-human.
2) were already pretty good at containing disease. When is the last serious epidemic in a first world country that you recall? The closest was AIDS, and we almost immediately adapted to better use of condoms and other protection, testing those with it, and creating treatments to allow those with HIV to survive. Our science took a disease that could have been a population decimating disease (look at what it's done in some parts of Africa, where they did not have the same access to technology and information as first world countries) and allowed us to curtail it's spread, and treat it's effects; and this is a disease we didn't have means of fighting directly, we stopped it simply by identifying how it spread and preventing that agent.
I'm not saying that diseases don't stop happening, HIV is still around and hardly trivial, but it's also no where near pandemic levels. The further you put us into the future the more informed we will be about disease and how they spread, the more capable we will be at creating vacines or even anti-virals to combat these diseases. Put us far enough in the future that we have such covenient space travel and we will likely be able to combat diseases with direct treatments for most new disease. However, even if we can't we can control them and keep their spread limited quite easily simply by indentifying how they spread and stopping that spread.
The only disease I see being a threat would be one that lied dormant for an extensive length of time, to allow spreading to a huge percentage of population before we identified it and started to fight it. However, disease doesn't do this, but it's very nature it will always spread as fast as it can, it's not 'smart' enough to know to lie dormant to spread that long. Perhaps more accurately, any disease that spread so successfully while dormant would not become lethal all at once, dormant spreading is proving an evolutionary advantage over killing the host, why should it then adapt to being lethal? (and yes, I'm overly personifying disease and their 'decision' to adapt, but I stand by the the raw arguments as true, even if I'm skipping over some of the specifics of unguided nature of evolution to provide a simpler explanation).
In short, we have means of stopping spread of diseases now; will have even better means of directly identifying and detecting diseases later, and they will only be a threat to one species at any time.
furthermore, the enviroment you hypothesis would make controlling disease easier. Each world is still mostly cut off from the other. Yes you said that travel was common, but each world is an isolated unit with very controlled 'borders' for people to come and go. If one world is infected it can be isolated much easier, and even if it's disease spreads you can identify likely worlds to qurintee easier. You can enact policies to screen people coming from known danterous worlds and since all travelers will have to come via ship that enter spacedocks that are controlled you have a very effective way of screening to ensure no one enters a new world with disease; as opposed to now when someone can walk across a border from a city to another in any of a million paths and bring disease with them.
Furthermore if aliens are really intermingled with humans you get a quasi herd-immunity as well. Since the aliens can't catch the disease they are immune. If worlds are culturally diverse then you may have only 10% of your population being made up of any one species, meaning 90% of the population is immune to disease. Your get at least some limited benefit of herd immunity this way.
The biggest danger would be not from diseases that live in and attack the human (or alien) body, but from outside containment that are universally dangerous. Radioactive materials, invasive species and 'parasites' that don't live in the body, but around it and are harmful (imagine space fleas that evolve to live in material commonly used by space ships and can damage instruments or the like)
Still, even in these cases qurtine techniques would work well, and more advanced science means better detectors for such threats.
Thus the only possible pandemic level threat I could see are man(or alien) made threats. Genetically engineered diseases or a nano-plauge. These diseases could be designed to be lethal, most likely by having them spread for years before they become lethal, so they have time to ensure everyone catches them. Nano-anything has a theoretically horrible grey goo worst case scenario, be it intentional or accidental. However, in the end I think man would have to be involved in creating anything that could prose a serious threat to a species.
Of course diseases will still exist, and spread. Pandemic level disease won't occur, minor diseases will always be around, even lethal diseases will spread to some degree, just being contained to a mall number. Once a disease reaches a certain danger level humans will invest effort into stopping it, and will; but diseases below a certain threshold won't be worth the effort to contain. Even contained disease will still take lives as well, look at HIV in the US now, it's 'controlled' and unlikely to be a serious threat to out population as a whole, that doesn't mean it doesn't take a number of lives.