You cannot deny that it opens up a larger, more difficult to control border for the European Union. This affects both security and immigration issues.
I have been to both Europe and Russia on multiple occasions, and have seen that the number of minorities are growing, particularly in asian populations, where the southern borders in the far east have been difficult to control, as China's population is overflowing along the border, with people looking to come over. Opening a border with Russia could only hurt those wanting to live in Europe from Asia, and from Russia itself.
As for unstable countries. One could argue that Georgia is not a stable country, with recent change in leadership and the ousting of Shevernadze. In addition they have had problems with autonomous republics. Also, Georgia has recently arrested nuclear material smugglers who were trying to sell parts from an Armenian power plant. In addition there have been similar arrests with authorities claiming that material either came from Ukraine or Russia. If you look up nuclear trafficking online, there are a whole bunch of examples, as people look to sell small radioactive materials that came from various generators or plants. This is not the kind of material that the EU would want coming across its borders. Not to say that it would, but they would be making it easier. Your argument will now be that the Georgians arrested these people, and that their security was sufficient, but there are still many former Soviet republics like Georgia that have nuclear material left over that is not in the most secure places.
In addition, one could argue that Russia is not a stable country in terms of its security. Do the same search and youll come across many of the same stories that have happened in Russia. The enforcement caught suspects in the cases mentioned, but the fact that these opportunities exist would be enough of a red flag.