While not 'the killers of today'¹, the common cold seemed to remain a problem until the mid-24th century.
In a rough progression of broadcast (not chronological and not complete):
ST:TOS Dr. Leonard McCoy
Doctor Leonard McCoy noted that modern medicine was still searching for a cure for the common cold in the 23rd century. (TOS: "Plato's Stepchildren") McCoy found a number of promising biological candidates on Omega IV that might lead to a viable cure. (TOS: "The Omega Glory") Scotty also mentioned that the common cold was yet uncured, comparing Mira Romaine's discomfort with space to this affliction. (TOS: "The Lights of Zetar")
By the mid-24th century, the common cold was apparently cured, though earlier in the century it still existed. (TNG: "The Battle", "Datalore", "Ensign Ro")
ST:ENT Various personalities
In 2151, Malcolm Reed was surprised that, although Humans could travel past light speed, they still could not cure the common cold. Phlox replied that there was no cure or treatment for the illness except for methods similar to those used in the 20th century. Reed wondered how he had caught the virus inside a hermetically sealed starship, causing Phlox to suggest that he might have picked it up from a contaminated tool or a sealed container. When Reed recalled that he had opened a case of plasma coolant, Phlox surmised the coolant's packer had been nursing a cold. He dismissed Reed's protests that it had happened five months ago at spacedock, saying viruses could survive and adapt. (ENT: "Sleeping Dogs")
ST:DIS Saru and Linus
In 2257, when Saru thought he had a cold, Linus told him that he had a cold the previous week, "which sucked!" as he remarked. (DIS: "An Obol for Charon")
See the full Memory Alpha - Common Cold page for more info. I think I've regurgitated enough of the important bits to comprise an acceptable answer.
¹ The CDC estimates that as many as 56,000 people die from the flu or flu-like illness each year.
Quoted narrative sources:
Memory Alpha - Common Cold