I agree with David.
Safety, or rather the "feeling of safety", is always a very subjective matter so these are always difficult questions to respond to... I personally wouldn't worry about this, for me, Carrer del Carme is perfectly all right and it's not more problematic, safety-wise, than other streets in El Gòtic quarter. However, and here it comes "perception", this is a narrow street with buildings 4 or 5 storeys high therefore a bit darker than other streets and for some people, this can "feel" a bit intimidating and/or less safe -which it's not! Note, however, this is something common in this part of the city, which is part of the cradle of Barcelona, the Old City. And as in all old cities anywhere else in Europe, there are always streets that are narrow, darker and a bit rough-edged...
If you meant to stay there... since you say you're "not used to big cities", why don't you stay somewhere in l'Eixample district instead? it's very centrally located, streets are wide and luminous and you'll probably feel much more at ease, especially if you return late to your hotel/apt. Note however that you still should stroll in Carrer de la Mercè and the surrounding area... no reason why not to.
Districts: http://bit.ly/BCNdistricts
Neighbourhoods: http://bit.ly/BCNhoods
If you meant to go there just for tapas... IMHO I'm afraid most of the bars in that particular street are not really worth going to, for me, there's life beyond the oily deep-fried fish and seafood served for example in Bar Celta or at Tasca el Corral. Albeit I can understand the allure of some for visiting some of those places... raw rascal and underworld-like, at the end of the day most of the tapas they serve are not my particular type of tapas. But again, that's me of course :)
Enjoy!