The purported etiology of infected clothes (the "smallpox blankets", but it was also done with yellow fever famously in the Civil War, and typhoid too) spreading disease is that they are fomites. This comes from contagion theory, not the theory of miasma. Fracastoro first purposed contagion as a mechanism for disease spread in the early 16th century. As you point out, the traditional explanation for epidemic disease was miasma -- bad air. The ancients believed that all disease was due to imbalances (or dyscrasias) in the four humors. This was as easy enough explanatory model for an individual -- but what about epidemics, when everyone in a town would get sick? Since the humors could easily be disrupted by environmental factors, and everyone knew that swamps were a major source of the daily quotidian fever (likely malaria), miasma was an elegant solution. When the Justinian plague ripped through the Mediterranean, it was due to some environmental swath of bad vapors.

Doctors in Italy in the 16th century were confronted with a new type of epidemic disease, however -- syphilis, brought back from the New World. This was the first recognized sexually transmitted infection (gonorrhea and chlamydia are much older, but had not clearly been linked to intercourse, and it actually took until the 19th century to prove that they were distinct from syphilis). Now you had a horrible disease very clearly linked to sexual contact -- and in fact, the initial lesion was at the point of contact. How could miasma explain that?

It couldn't, and Fracastoro proposed a new idea of epidemic disease -- contagion, or by touch. There were vigorous debates in epidemics of the mechanism, with important implications. If you believed in contagion, you would isolate, keep infected people away from others. Miasma, you cleaned the streets, handed out clean blankets, &c. The miasma versus contagion debate raged up until the 1850s, and played a major part in the response to cholera.

Anyway, the British and English-speaking world easily accepted that smallpox was caused, in fact, by contagion and not miasma (as it had earlier). Unfortunately, I can't speak as much for physicians in the rest of Europe, though vaccination was taken up enthusiastically throughout Europe. And the reason for this is largely variolation -- the inoculation of smallpox pustules into the arm to protect the individual with a less severe infection. Unlike, say, cholera, there was a clear link to exposure.

I should also add that many attempts to use fomites would have failed. Yellow fever was used in the American Civil War -- but of course we know today that yellow fever is spread by mosquitoes, not fomites. It's unclear how effective smallpox blankets would have been; it turns out that transporting smallpox is a lot more difficult than you'd think. There's only one confirmed case of smallpox being used as a weapon in war, but strong circumstantial evidence of a second using variolation material.

Finally, miasma theory was not around in the early 20th century. Germ theory put the final nail in the coffin, but growing epidemiological data collected during the second cholera outbreak, and with yellow fever in the Caribbean and American south, had been slowly destroying the intellectual basis of the theory throughout the second half of the nineteenth century. By the 1880s, you probably wouldn't find any doctors in Europe who believed in miasma, though it took a decade longer in the United States.