The only downside I see to mixing tires, is the potential that the tires could have different performance abilities.  If one tire is poor in the wet, then that end of the bike may not be able to handle the wet weather as well....but is one loose end worse than having 2 ends sliding around?

As tires age they become harder, and they lose their ability to grip in turns.  Wet weather is the place where it is most noticeable.  Also the harder rubber tends to cup or wear unevenly, and the ride quality suffers.  The little Ninja 250 I bought last year had the original 11 year old tires and 9,800 miles on them, and when I put on new tires the ride and handling really improved.

If you do a search on motorcycle tire recommendations, they claim you should check the tire code and never mount a tire that is more than 3 years old, and replace your tires when they are more than 5 years old.  You should also plan on buying tires that will provide no more than 3 years worth of riding for you....there is no benefit to buying Michelin Commander tires if you only ride 1,000 miles a year - as the rubber will turn hard long before you ever wear the tread off.  The 2002 Savage I bought this year has the original tires on it with 4,000 miles on them and they look great, and if I rode at a moderate and mild pace they would be just fine - but if you push them you soon discover they are hard and don't grip the pavement well.  My Chevy Avalance has tires on it that were made in 2006, and they are downright scary when the roads are wet....they slide all over the place.