As far as I could ever tell, as a frequent visitor in the very late Soviet period, mainly to Moscow (apologies to our esteemed poster Tim, whom this seems to annoy for some reason), it wasn't anywhere near as bad. Moscow was still a very safe place for everybody, and I felt much more secure wandering about in the early hours, or at any other time of day, than I did in my native Manchester. If I hadn't been white and easily able to blend in I'm not sure however.
Sure enough, the official line was that all peoples are equally deserving of respect, but I do have some uncomfortable memories. For instance, I met several people on the trains from Ostend to Moscow from Africa and the Caribbean who were studying or working in Moscow. On one occasion me and some Ukrainians I'd got talking to (Russian speakers like most Ukrainians) from Lvov were drinking together in a compartment, and when this young Jamaican on his way back to Moscow for the new academic year I'd earlier got talking to passed by they invited him in and happily tried to make conversation with him, showing friendly interest in what took him to Moscow, as well as life in his homeland etc etc. A few years later I visited these people in Lvov (by then Lviv...), and they told me that they were so incensed by his having a Russian girlfriend (with whom he'd had a child) that they'd have kicked his teeth in given the opportunity. They were not 'political' in any way I could tell, just ordinary lads with wives and kids.
Shortly after the 1991 attempted coup I also met an African male, travelling to London, where he was planning to settle if possible as he quite wisely surmised that the end of the USSR was going to see an upsurge in overt racism. He'd been in the USSR since 1980 and had settled in Moldova, had an ethnic Ukrainian wife and several kids, but the marriage had ended along with his employment. He did somehow manage to settle in the UK (precariously it seemed), and when he came up to see me about a year later it was while my girlfriend (loosely termed as you can't really have a stable relationship with somebody at the far end of the continent) from Moscow was over on her first visit. He had reason to make a trip to Moldova some time in the near future, and as he'd have to go via Moscow was sounding us out about places to stay. I noticed how uncomfortable she was about the idea that she put him up, and later she said to me something along the lines of 'What do you think my neighbours would say if a black man turned up at my door?' She was your typical chattering class Moscow 'liberal.'
While renewing my visa one time, at an office on Moscow's Rochdale Street, named in honour of the town where the co-operative movement was formed, I found my Russian to be inadequate to filling in the complicated form and so approached some Africans as the most likely people to speak English in the chaotic, crowded office, to compare notes. I didn't get the impression that they were being treated particularly well by the impatient, harrassed officials, but neither was I.
In 1988, on my first trip to Russia, I went via Poland, and having lost my way wandering about in Warsaw I approached an African, again as the most likely English speaker. He was studying there and had been in the USSR. When I told him I was on my way to the SU he said 'It will be a strange experience but you will meet some really nice people...'
So all in all a mixed picture.