A formerly influential LiveJournal blogger, Alexey Trankov, wrote [ru] the other day that judging by the amount of people protesting on the streets in Ukraine, as compared with the number of people discussing the protests on the Russian Internet, Russians seemingly care more about what happens in Kiev than Ukrainians themselves. Although a bit of a hyperbole (estimates of Sunday's march and rally in Kiev are as high as 100,000  [ru]), Russians do care about what happens to their next door neighbor quite a bit.
Not that there is any consensus in Russia on whether Ukraine should continue the long process of EU accession — the demand made by the protesters — or whether it should forge yet closer ties with Russia. The situation is complicated by the regional subdivisions of Ukraine itself, illustrated on this map posted by Pavel Pryanikov of ttolk.ru:
The three regions, Ukrainian-speaking west, Russian speaking East, and the Russo-Ukrainian creole center, generally coincide with the preferences of the inhabitants — the East is pro-Russia, the west is pro-West.
In Russia the divisions are ideological rather than geographic. Leftist writer and politician Eduard Limonov noted that Russia's liberals vicariously support Ukrainian protesters:
Ультра-либералы РФ возбудились майданом больше украинцев.Свой майдан передвинули, предали и профукали,а от чужого готовы в пляс пуститься.
— Эдуард Лимонов (@edvlimonov) November 24, 2013 
Russian ultra-liberals are more excited by Maidan than the Ukrainians. They've moved, betrayed and wasted their own Maidan, but when they see a foreign one, they are ready to dance.
Some of these liberal activists, like Roman Dobrokhotov, fondly remembered the Orange Revolution:
Как и в 2004-м, украинцы дарят праздник моему сердцу. На фоне средневекового гнилья, которое взбаламутилось в РФ- это глоток свежего воздуха — Roman Dobrokhotov (@Dobrokhotov) November 24, 2013 
As in 2004 Ukrainians have gladdened my heart. Compared to the medieval rot that has been stirred up in Russia, this is a breath of fresh air
Others, like journalist Arkadiy Babchenko, made comparisons  [ru] to Russian Bolotnaya Square protests, which some contend were sabotaged by collaboration with the government:
[…] не повторяйте наших ошибок. Никаких согласованных митингов. […] У нас не получилось – может, у вас получится.
[…] don't repeat our mistakes. Don't have any sanctioned rallies. […] We couldn't do it, maybe you will.
Other liberals were more ambivalent, like Oleg Kozyrev, usually a protest cheerleader, who wrote:
Сейчас в Киеве решается, за чей счет будет жить Украина – за счет пенсий российских стариков или за счет пенсий немецких стариков
— Олег Козырев (@oleg_kozyrev) November 25, 2013 
What's being decided in Kiev right now is who will pay for Ukraine — Russian pensioners or German pensioners.
Conservative publicist Dmitry Bavyrin, on the other hand, took a realpolitik view  [ru] of Ukrainian unrest. According to him, there is no reason that any Russians should support Ukraine's joining the EU, since it would be a geopolitical and economic loss for Russia. He also criticized those members of the opposition that support Ukrainian protesters and thus discredit the Russian democratic movement:
с обывательской точки зрения, нет ни одной причины для того, чтобы российские политики поддерживали украинскую евробучу. Ни одной, ни малейшей. Только если не предположить, что за это платят условные “хозяева из Госдепа”. И эту мысль госпропаганда до обывателя очень скоро донесет.
from the point of view of the man on the street, there isn't a single reason for Russian politicians to support the Ukrainian euro-fight. Not one, not the slightest. Only if you conjecture that it's being paid for by abstract “bosses from the State Dept.” And this thought will be brought home to the man on the street by the state propaganda machine very soon.
Meanwhile, a Ukrainian blogger writing in Russian, Aleksandr Volodarsky, thinks [ru] that the entire situation is likely much ado about nothing:
Возможный сценарий — Янукович заигрывает с Россией до 2015-го года, а потом внезапно сближается с ЕС перед выборами. […] Если провести ассоциацию сейчас, то за два года люди успеют понять, что никаких бонусов от этого они не получили: небо не стало голубее, воздух не стал чище, границы закрыты, в европейских посольствах хамят как и раньше. И возропщут.
One possible scenario — [President] Yanukovich flirts with Russia until 2015, and then suddenly gets close to the EU before elections. […] If they put the association through right now, then in two years people will become aware that they didn't get any bonuses from this: the sky isn't bluer, the air isn't cleaner, borders are closed, the European embassies are as rude as ever. And then there'll be unrest.