Is there a danger of the plastic bag melting for cooking via sous vide?
How do you know when the plastic has melted?
Polyethylene, which is the material typically used for sous vide bags, is stable at the temperatures the sous vide bath will reach. Ensuring you're using the right kind of bag, and checking that the temperature hasn't gone too high, should avoid any risk of melting.
Polyethelene is much more stable than plasticized plastics. It's fine at the low temperatures typical of sous-vide cooking (which generally don't creep above 140 degrees Fahrenheit). They do, however, start to soften at temperatures above 176 degrees Fahrenheit...
According to the latest research, the safest plastics for use with food are high-density polyethylene, low-density polyethylene, and polypropylene. Virtually all sous vide bags are made from these plastics, as are most brand-name food storage bags and plastic wraps such as Saran wrap.... The bottom line is that bags made expressly for cooking sous vide are perfectly safe as are oven bags, popular brands of zip-top bags, and stretchy plastics such as Saran wrap.
See also Sous vide without plastic bags? for some discussion of plastic bag safety and alternatives.