How can I infuse alcohol with 'fragile' herbs?
Refrain from muddling, use more (fresh) basil, (try agitating,) and infuse for a longer duration; I quickly found one recipe for basil-infused vodka requiring a four day sit and two "fists" of basil. Vanilla and ginger can take a week, cucumber vodkas can entail a fourteen day sit.


Some constituents will infuse quickly (habanero, strong flavor), some will not (cucumber, subtle flavor) 
Some should be processed whole to avoid "off-flavors" or other reactions like browning (basil), while others should be crushed (garlic, promoting allicin production), shredded (carrots, increasing surface to area to increase extraction), or chopped (pears).


Many variables play into how various ingredients will infuse various liquors, and I don't feel it would be fair to really say there is a general rule to follow. To begin, though, I would identify your main variables with respect to solute (look for basil recipes, if none then other leafy herbs), solvent (look at the pH of the alcohol in addition to the type and proof), and method (duration, preparation of solute, agitations, single v. multiple/compound macerations), and look to similar approaches.

Hands-on Patience is the real virtue here though; vis-a-vis frequent sampling. From everything I have read and seen in both my limited experience and another friend's, even returning to what you thought was a 'tried-and-true' can yield very different results. 


With respect to discussions on vodka &amp; gin VS. Everclear extractions, there is a Chow thread of interest
With respect to the industrial processing of macerations and compound distillates, I found an interesting article on ICS
If the browning continues, you might consider adding a citrus like lime or lemon to see if that stems the browning as the added acid may inhibit this (Gordon's has a pH of 6.9, others, particularly cheaper gins, range down to 3.8). That said, in my limited explorations with infused vodkas I have not run into a browning issue so it may simply result from muddling (bruising).