On request by <PERSON>:
This question is actually a lot harder as it appears on first glance. If you innoculate the population of the Americas you merely put back the disaster one generation (or less depending on how long the immunity from inoculation is actually for). So, you have to be willing to keep it up in the new world forever (they'll be no push to bring the live vaccines over if the diseases mysteriously aren't appearing) or fix it another way.
I have considered two ways. You could in theory force the disease breakouts centuries earlier with the same catastrophic kill rates, but this is ethically unsound and significantly more dangerous due to increasing the genetic bottleneck that made the natives uniform enough to have one blood type.
The other solution is to kill off the deadly diseases in time to prevent the disaster. You will need multiple trips.
1. Pick a year. 1492 will do. In fact this is a very good time as the Portuguese have started sailing around Africa.
2. Gather samples of smallpox, measles, mumps, and greatpox (the old name for syphilis).
3. Extract the genes for their surface proteins (DNA or RNA)
4. Transplant the genes into a modern copy of the common cold along with activation sequences.
5. 335
Test your new viruses carefully. There's a small chance you made an airborne killer.
6. Release these viruses at multiple points in the old world all at once. The greatpox one is to be used in the new world.
7. Repeat as necessary near any residual breakouts. You have time travel that permits editing the past: you can watch the news and see where any breakouts come from and try again to stop them.
If you did your job right you have modified the common cold into an airborne vaccine or set of vaccines against the killing diseases. The genes you spliced into the cold virus aren't doing it any good so they'll be withered away sooner or later but this doesn't have to last all that long. Just long enough.
Presto. Epidemics of the Colombian Exchange halted. If anybody learns you did it they will proclaim you a hero.
So, if you're wondering how this works: The transplanted surface protein won't do much in the situation. It's either going to be incorporated into the cold's surface protein matrix to be picked up by the B-cells for antibody manufacture (we want this) or made but not picked up and eventually dumped into the intracellular matrix (almost as good). So long as we weren't silly enough to use the site-latching surface protein from the target virus that would be picked up by the common cold virus the resulting virus is nearly as safe as a cold (small chance of autoimmune disease from mumps). 225