How should I deal with blood released while thawing meat in the refrigerator?
To start with, the red, or dark, juice from red meat is not, in fact, blood, which is a common misconception. Most blood is drained from red meat when it is butchered. It is, rather, a protein (myoglobin) and a lot of water.

It is an animal's levels (or lack of) myoglobin, that determine whether it is a 'red' meat or white.

As for its safety after being frozen, this is discussed further here.


  Beef stored in the refrigerator for more than 5 days will start to turn brown due to chemical changes in the myoglobin.  This doesn’t necessarily mean it has gone bad, though with this length of unfrozen storage, it may have.  Best to use your nose to tell for sure, not your eyes


As your venison was frozen, rather than just refrigerated, as mentioned above, I would say you are more than safe to do a nose test, as recommended above.