Evolve usually has a positive connotation. Evolution is usually thought of as making improvements or refinements. The Latin meaning of e-, “coming out”, is mostly forgotten, but lives on in evolve’s sense of “gradually realize a more perfect form”. Refinements can make a bad thing stronger, of course, like when a virus evolves or when an evil genius evolves a plan for world domination. The positive connotation derives only from the denotation of refinement or fulfillment; if something negative becomes more refined, that’s worse, of course.
Evolve has a neutral connotation when it means mere change or the unfolding of some process without any suggestion of refinement. The “coming out” meaning of e- also appears in the sense of “gradually become apparent”, which carries no positive or negative connotation of its own, though it might take on the tone of whatever became apparent.
Devolve is the opposite of “evolve”: to become less refined or less organized. This word might have exactly the negative meaning you’re looking for.
Devolve has other meanings, too, like moving decision-making authority from higher in a hierarchy to lower in that hierarchy. The de- still carries its Latin meaning of “coming down” fairly clearly. Whether the connotation is positive or negative depends on what sort of coming down you’re talking about, of course.
The spare motive that begins the sonata evolves into a long, lyrical melody by the coda. [positive connotation]
Today’s software-engineering practices have evolved from about 50 years of practical experience. [positive connotation]
The protest has devolved into a riot. [negative meaning]
The protest has evolved into a riot. [neutral connotation, describing something bad, suggesting only that the change happened gradually]
It has evolved that the children have been using stackexchange.com to do their homework. [negative, only because of what is described]