I'm assuming you made a pseudo soft-drink Ginger Beer, where it's mostly a sweet beverage, and yeast is primarily used to bring carbonation (and only incidental alcohol).
The problem you will have is over-carbonation, the ginger beer will not "go bad". The yeast will continue consuming the sugar until none is left, or the amount of alcohol it produces is self-toxic. Bread making yeast will typically get to about 14% AbV before this happens, wild yeast (like from a "ginger bug") probably somewhat less, and Champagne yeast ~ 19%. If you dislike alcohol, and a less-sweet flavour, then sure, it's "bad". But during fermentation the yeast is creating alcohol, lowering the pH, and removing oxygen. This makes a fermented beverage fairly inhospitable to other micro-organisms. It will keep well.
The big issue is whether the vessel the ginger beer is sealed in can take the excess carbon dioxide pressure. If the bottles are glass, then this is a dangerous situation. You can mitigate this by periodically venting the excess gas from the bottles. But if the bottles are already gushing out, then it's a problem already - if your bottles are glass be really, really careful (at least wear eye protection). With PET bottles, the risk of flying bottle-shards is mitigated.
The yeast will not stop in the refrigerator, just slow down a little. Freezing? Freezing should stop fermentation, but it will not kill the yeast. You could try bottle pasteurisation and perhaps preservatives (like potassium sorbate, etc.) to stop fermentation - use techniques as if you were making apple cider.
If you like the taste of the yeast, it's no problem to mix it up into the beverage. But for a clearer product, all you can do is cool it down, and wait for the yeast to settle out - then pour carefully.
