I know there's this big pile of "you must replace your struts" replies.
But make sure that they're really "broken" first. Take the car to your best, local, independent front end/alignment shop. Don't say anything about struts. Just ask them to inspect your suspension. If they also come back saying that you need struts (remember not to plant that ahead of time), then replace your struts.
I drive a '97 Escort. Granted that it spends most of its time on reasonably well kept interstate highways, but I put over 200K on the front struts. When I replaced them it was because the strut bearings were binding - not because I had any problem with the struts. Based on the fact that the ride and handling didn't change at all I concluded that those original struts were still just fine.
The rear springs on those cars are a weak point. I replace my rear struts at about 150K miles - but the struts themselves were fine. I did just have a broken spring.
Now - since you could be confused you might clarify with them exactly what they claim is broken. If they are talking about springs then you absolutely cannot drive on broken springs.
(I don't know if this is confusing but the strut itself is just one part of a whole assembly of things - look at the diagram in the middle of this page: http://www.autopartslib.com/2010/03/18/car/2005-acura-tsx-front-suspension-strut-assembly-parts-schematic-diagram - it says its for an Acura but a strut assembly is just strut assembly).
In any case, just get a trustworthy second opinion.