They survived with the medicine that was available the same way people have done so for millennia: namely, some survived and some did not (as is the case today). The small pox vaccine was available, of course, but it was not as widely used as it might have been. There were, consequently, small pox outbreaks in the West during the nineteenth century. Death records document those who did not survive, but they only hint at the number of people who contracted the virus and lived.
One of the problems with the medical science of the time was the lack of a reliable needle to inject vaccines. The small pox vaccine was administered with scratches on the skin: like everyone of my generation (and like younger people in the US military who were sent to the Middle East), I still have my small pox vaccination scar.
One of our excavations in Virginia City, Nevada uncovered several hypodermic devices, but they could not be called "needles" by any modern sense of the term. They ended in a rubber "point" with an opening of about 1 mm. They were used to administer drugs. Unfortunately, tests were not able to determine what drugs were present, but we assumed that they deployed opiates. We were able to do DNA testing. The procedure for using the device was to open a vein and then to work the end of the hypodermic into the wound, irrigating it so the drug would be taken into the blood stream. The process consequently would collect fatty and other tissues into the opening of the device, which opened the door for DNA testing: we found that two men and two women had used the device. This a good argument for anyone who discovers how to travel in time to be prepared to return immediately if medical care is needed!
For the most part, diseases that now warrant vaccination were simply endured. There were casualties and there was discomfort and other consequences, but while life can be fragile, it can also be tenacious. Try as the world might, humanity persists.