It definitely goes back further than 1981! The simplest answer is that people had a lot of sympathy for them because they had been working on building up sympathy before the Revolution.

During the nineteenth century, royal families across Europe built up the kind of monarchy celebrity culture we know today. They had been losing their ability to seriously control their countries, particularly in the wake of republican revolutions, and had to rely more on influence than sheer power - influence they could wield with their ministers, and influence they could wield with public opinion. For instance, Queen Victoria made the British royal family a pattern for domesticity in a time where a woman's ability to make a welcoming home for her husband and raise her children with high moral standards was paramount, which gave her opinions more weight with her subjects. She and other crowned heads would have photos taken with their spouses and family groups in normal clothes for public dissemination, showing them as individuals just like the viewers. (Once movie reels were invented, royal families took advantage of them as well.) Families were key, because heirs really had nothing much to do and could be sent to do public charity work or have their marriages turned into fairy tales.

The Romanovs engaged in this like the others, but only to an extent. Nikolai and Alexandra, rather like Louis XVI and Marie Antoinette, were not interested in the performative aspect of monarchy and preferred to live privately. (Alexandra in particular had grown up in a very non-performative setting in the small German state of Hesse, the farthest thing from Russia, the last holdout of the old autocracies.) They had the expected opulent fairy-tale state wedding at the Winter Palace in 1895 at the behest of the rest of the royal family in line with protocol, but immediately afterward they retired to private rooms, and for the rest of their marriage they would make every effort to avoid the public eye - and even when they were to be found in public, unrest and assassinations required an intense level of security around them. However, they still engaged a royal photographer/cinematographer in order to keep up with the expected use of the media, and so even while the Russian people rarely saw their royal family in person, they were very familiar with the image of the handsome parents, pretty and well-dressed daughters, and little son. When World War I broke out, Alexandra and the two oldest daughters (Olga and Tatiana) trained with the Red Cross to be nurses; Alexandra took over the official relief effort and converted the Winter Palace into a giant hospital, and photographs of the three in uniform were frequently used somewhat propagandistically to exhort others to join up as nurses, roll bandages, donate money, etc. Olga and Tatiana went to work like any other nurses at an annex of the Court Hospital, where Alexandra required that they not be treated as princesses. They also traveled around the country to visit and inspect hospitals, often with the three younger children as well. All in all, the war gave them a huge amount of positive press and gave a huge number of people positive personal interactions with them.

So by the time of the Revolution the Romanovs were not universally hated. If they had been, they would probably have been dealt with more publicly - put in a state prison, perhaps taken out for the people to see under guard and humiliated, and executed with witnesses. Instead, they were essentially hidden in Siberia and executed along with their servants in a basement, and once they were killed there was no official announcement. In fact, the government told people that Nikolai had been executed and the others moved on to another location, refusing to confirm that the entire family had died for some time, even after the pretenders started to appear; the first reports that they had all been killed came about from an inspection done under the White Army in 1919 when they captured Yekaterinberg.

Many people hated the tsar, and with good reason. Many people also hated the tsarina, whose health problems were as hushed-up as the tsarevitch's and who therefore appeared to have no excuse for the many times she was absent from court functions or appeared unhappy in public; like Marie Antoinette again, she was also disliked by many members of her own court as an "outsider" who didn't conform well to the expected etiquette. But their fourteen year old son? Their daughters, aged twenty-two to seventeen? Is it really surprising that people would view this as a brutal slaughter and a tragedy rather than a justified execution? The children had no place in the political structure, and even Alexandra had had only a modicum of actual power (when she acted briefly as regent in Nikolai's absence during the war), so it simply did not occur to most people to hold them as responsible as their father/husband for his mismanagement and cruelty.

While my knowledge concerns more with the foreign policy and revolutionary elements of Russia, I think I can maybe offer a little in way of an answer. Nicholas is a bit of a bizarre tsar. You accurately point out that he was quite reactionary and anti-semitic, however, I believe in order to truly understand Nicholas, one needs to understand the context. Nicholas inherited a largely archaic and arguably broken empire. His grandfather, Alexander II, partially liberalized the country but truly only opened festering wounds. His liberalization program, while bold for a Russian autocrat, never fully addressed the issues inherent in Russia. Ultimately, the very people that Alexander, to a certain extent, tried to help assassinated him brutally. In comes Alexander III, who never was supposed to be tsar (his older brother, who was trained to be tsar, died in a bizarre accident). Alexander III was never very intelligent and always had conservative leanings, which elements of the government helped foster. It did not help that Alexander III saw his horribly disfigured father on his deathbed. Consequently, Alexander III went about a very conservative and reactionary policy. He cracked down on dissent and terrorism, while simultaneously silencing many liberal voices. He also isolated his family from Russian life in an attempt to avoid assassination (there were still a few attempts). Thus, Nicholas was raised in a very isolated and deeply conservative household.

Unfortunately for Nicholas, his father died relatively young leaving him with a very divided and angry empire. Alexander II's reforms opened the country just enough to create chaos and Alexander III's policies merely threw gas on the fire. Nicholas was not trained to be Tsar, so his policies largely followed his father's. Nicholas, raised to believe in the autocracy, never once believed in reform. If anything he wanted to go back to an almost medieval version of Russia. However, the country was moving towards reform. 1904-1905 was the first blow to Nicholas' understanding of Russia. His people rebelled and demanded a constitutional government. Nicholas only begrudgingly agreed to the Duma (basically Russia's congress) and then immediately ignored it. Representation and democracy were antithetical to Nicholas' conception of his empire and he could not truly accept the modern world.

Now to get back to why historians tend to look sympathetically back on Nicholas, it boils down to the fact that Nicholas was simply an ill-equipped man for the time. He was a good family man and deeply cared for his children. He was very religious and considered himself a good Christian. The problems he had to contend with were simply too great for him to bear. It's unlikely even someone of Peter the Great's ability could continue the autocracy of Russia. Nicholas was merely the wrong man for the wrong time. Then to further soften the image of Nicholas, one must realize how brutally he was killed. His family and himself were dragged to a basement and gunned down. His body was unceremoniously disposed of and left to be forgotten for decades (on a sidenote Yeltsin was briefly in charge of the city where the bodies resided and kept the knowledge of the killings secret).

In terms of sources, there are a few good ones to look at. Simon Sebag Montefiore has a good book on the Romanovs appropriately titles The Romanovs. For a slightly more outside perspective on Nicholas, Paul Robinson's Grand Duke Nikolai Nikolaevich is a good book (the Grand Duke Nikolai is a cousin of Nicholas II. It can get a bit confusing). Finally, I strongly recommend Phillip Blom's and Veronica Buckley's Twilight of the Romanovs. It is a compilation of various photographs from the era of Nicholas II including some of his family. It very much humanizes Nicholas.

To sum up as best as possible, Nicholas was a good man (at least in terms of a Russian in 1917) but a horrible tsar. Historians largely pity him simply because there was nothing he could do. He and his family were autocrats to the bone. Russia however was modernizing and Nicholas either had to step down or be caught up in events. His life was simply a Greek tragedy.