How long will a fresh herb paste last?
Well of all your listed ingredients the only two that can really 'spoil' are the fresh basil and fresh mint. Honey never goes bad, and nearly any dried spice can last for a long time with just a degradation in flavor.

I'd say that if you washed and dried (sanitarily) your basil and mint before addition to this mix that you could get at least 1-2 weeks out of it unrefrigerated, and at least 1-2 months out of it refrigerated.

Please note that this is just a ballpark estimate and you may have to do some trial and error to determine a more specific window. That said, I wouldn't be surprised if you could get a significantly longer shelf life out of it than my estimates.

Update - I think the presence of salt does enough to "extend" the life of it. 

Another Update - Some Googling has informed that honey is in-fact a preservative. Apparently honey is a rather hostile environment for bacteria due to the production/presence of hydrogen peroxide.

I still think my estimates are accurate, albeit a little conservative now. The best thing you can do is trial and error.

Sources:

http://www.entrepreneur.com/tradejournals/article/132772675.html

http://natural-products.suite101.com/article.cfm/honey_as_a_natural_preservative