I wish there was a simple answer here, but the lack of clear science says: Maybe.  It's always tempting to equate causation with correlation, but just
because two things seem to occur in tandem — in this case, increased
incidence of melanoma among users of sunscreen — doesn't mean that one
causes the other.  Indeed, there are plenty of other possible explanations: People using sunscreen are more likely to be sun sensitive or spend more time in the sun to begin with.  Improper use of sunscreen may put this at-risk population at greater risk because they think they're protected, when they're not.  Sunscreens are much more effective at blocking UVB rays, which cause burning, than UVA, which causes cancer — limiting the natural "I should get out of the sun response."  That said, there is increasing concern about two widely-used sunscreen ingredients: Oxybenzone, a sun filter which can create free radicals, and retinyl palmitate, a form of vitamin A which one study has shown to accelerate cancer growth in lab animals.  There are plenty of places you can read detailed
discussions of the science involved in this debate — but I'm going to
focus on some practical recommendations: I'm not a doctor, or your mother — but here's what makes sense to me:

1. Limit your exposure!  This means hats, umbrellas and simple avoidance of being outside during the peak sun hours. 

2. Use enough sunscreen! Read the labels, people — it takes an ounce (think shot glass) of
most sunscreen products to cover an average adult. Chemical sunscreens (as opposed to physical sunscreens mentioned below)
need to be applied 20 minutes before you head outside. No matter how high the supposed SPF, you still need to reapply — every 60-80 minutes, after swimming, sweating or toweling off.  There's no such thing as "all day sunscreen."  If you're using a spray — be sure to thoroughly cover yourself — you'll need to get out of the wind to do it.

3. Look for sunscreens with physical sun filters — Titanium Dioxide and Zinc Oxide.  Most of these use nano-particle size ingredients — so you don't look like a white-faced kabuki dancer — and while there was some concern that these nano-particles are theoretically small enough to be absorbed by the skin, the current thinking is that's a more acceptable risk than chemical sun filters. (Finding non-nano Zinc and Titanium is nearly impossible).  I like  Jason Natural Sunbrellas, which rates really well in the Environmental
Working Group's
annual guide to sunscreen.

4. If you're traveling abroad, buy sunscreens with Tinosorb S, still not yet approved in the US.  The FDA has been dragging its heels in issuing new sunscreen guidelines and approving new products, no doubt in part due to lobbying by the competitive giant manufacturers, who have potentially have much to lose when new guidelines are approved.  If you buy sunscreen abroad or through a gray-market importer, it can be pricey — and treating it as precious and not using enough of them is again, not a good solution.