Problems:

Plastics contain many toxic chemicals that get either consumed by fish and eventually work their way through the food chain to humans, or wash onto shorelines, which humans are also in frequent contact with
Process of Degradation:
It is hard to say exactly how long it takes for plastic to degrade. In places like landfills, where the trash receives little sunlight, it can take 1000+ years. On the ocean floor in particular, which receives huge amounts of sunlight daily, they can degrade within a year. The only real way to degrade plastics is "photodegradation", where UV rays break up plastic molecules
Alternatives:
A plant-based hydro-biodegradable plastic. PLA decomposes into water and carbon dioxide in 47 to 90 days -- four times faster than a PET-based bag floating in the ocean