To be clear, Dr. Crusher never says they eradicated the common cold. In "The Battle", she says:
It may be true that headaches were once quite common, but this was in the days before the brain was charted, before we understood the nature of pain, when we were suffering from such things as the common cold.
But in "Datalore", Wesley Crusher and Data have the following exchange:
Wesley: Have you got a cold?
Data: A cold what?
Wesley: It's a disease my mother says people used to get.
Which does provide some circumstantial evidence that if it wasn't eradicated, it was certainly uncommon enough by the time Wesley was around to be something he knew nothing about except to parrot what his mother told him at one point.
While there is no canon specification for when the common cold was eradicated (or cured), one can formulate a conjecture based on circumstantial data like Wesley's recollection.
Here's the evidence in rough chronological order of what they describe:
As Memory Alpha notes, Dr. McCoy was still searching for a cure in the episode "Plato's Stepchildren".
Then, in "Ensign Ro", there's Picard's recollection of ginger tea being his "Aunt Rebecca's cure for the common cold."
Memory Alpha also refers to Tom Paris who, in the episode "Cathexis", tells the Doctor he had the common cold when he was nine years old.
Finally, there's Wesley's recollection in "Datalore" mentioned above
So, given these data points, I think it's safe to say the common cold was cured and/or eradicated in all but name sometime between the time Tom Paris was nine years old and when Wesley Crusher was born, or at least early childhood.
The year Tom Paris was age 9 is pretty tricky: his age is not mentioned in canon, so we don't really know. We could speculate that Tom Paris was the same age as his actor, Robert Duncan McNeill. Wikipedia says he was born in 1964, which would make him 30 years old when "Cathexis" aired.
Memory Alpha states that the events of that episode takes place in 2371, so if Tom Paris is the same age as McNeill, he would've been 9 in 2350.
Now, Memory Alpha states Wesley Crusher was born in 2348. It's unreferenced, but it's a good enough guess and likely comes from someone mentioning his age somewhere in the series, so I'd trust it.
Since Paris mentions having the cold after Wesley's birth year, it's hard to say when exactly it could've been eradicated. If there aren't any continuity errors, I'd wager it happened sometime between 2351 and 2353, which is when Wesley was 5 and likely would've already started to learn about the world in a substantial fashion from his mother.
But, on the other hand, it's likely there are continuity errors, and the writers didn't really expect to put too fine a point on it other than to trot it out any time they wanted to demonstrate how "advanced" Starfleet medical science was.