It's true that "minor arts" or "lesser arts" is used in a number of places to refer to this, but it is obviously a calque of the Spanish term. Although "artes menores" is quite specific, in that refers to a highly regulated sector in Spain with very specific rules on size of boat, type of tackle, methods, etc, it's basically the Spanish version of a phenomenon found internationally in the fishing industry, and specifically in the Mediterranean. The French call it "petits métiers", the Italians "piccola pesca". I think we could use the generic term "small-scale fishing", which is standard at European Level:

"Small-scale fisheries in the Common Fisheries Policy
[...] In a broader fisheries policy context, SSF may thereby be generally understood as referring to "small" vessels, but not necessarily excluding trawlers. The term "small-scale" is also quite often referred to as, or used in combination with, "coastal", "artisanal" or "traditional". Though not equivalent in practice, these denominations illustrate some of the characteristics frequently associated with SSF activities."
http://www.europarl.europa.eu/RegData/bibliotheque/briefing/...

Artes menores is really the latter kind, so you could add one of the additional terms suggested in parentheses in my answer.

Other highly relevant EU docs here:

General Fisheries Commission for the Mediterranean
Integration of Small-Scale Fishing in MPAs
See esp. "Profile of Mediterranean small-scale fisheries", p. 24 ff.
http://www.ssfsymposium.org/Documents/FullVersion/BPIII.pdf

And this one's good, with lots of pictures, and it's by a Spanish author:

"Small scale coastal fisheries under the CFP reform
Brussels, 25 February 2010
Casto López Benítez
Directorate General for Fisheries and Maritime Affairs
Facts and figures about SSCF"
http://ec.europa.eu/fisheries/news_and_events/events/worksho...

"Arts" is not at all satisfactory, really, since the meaning of arte here is "Instrumento que sirve para pescar" (def. 6 in DRAE). So you should really say "minor tackle" rather than "minor arts", as telefpro has indicated. But "minor tackle fishing" is neither explanatory nor idiomatic. I'd go for "small-scale". In order to reflect the specificity of the Spanish term, you could perhaps retain it on first mention, maybe even with a note if it seems necessary.

An alternative I considered is "artisan fishing", which is a recognised term, and the Spanish would certainly like to claim that "artes menores" means that, but I think it would convey the wrong impression; it's not industrial trawlers, but it's a bit more sophisticated that what is normally understood as "artisan fishing", in my opinion.

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Note added at 10 hrs (2014-10-14 15:11:39 GMT)
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Great reference! As you say, that nails it. It confirms my gut feeling that this is not really artisan; it is commercial, after all.