This article in Politico quotes some Senate Democrats as saying that Obama shouldn't even attempt to garner bipartisan support for any nominee:
“I think we need to push someone who would be on the liberal side, on the progressive side, just as Roberts and Alito are on that side,” said Sen. Tom Harkin (D-Iowa), referring to Chief Justice John Roberts and Associate Justice Samuel Alito, who were both confirmed in George W. Bush’s administration. “Why do conservatives always get the conservatives, but we don’t get to get liberals? What the hell is that all about?”
Not everyone agrees with Harkin's historical analysis.
On the Power Line blog, Scott Johnson quotes from an article by Stuart Taylor, about why conservatives hardly ever get the conservatives:
One reason why so many Republican appointees have turned out to be more liberal than the presidents who picked them has been the difficulty of getting nominees with conservative paper trails through the Senate.
The point is that many justices who were appointed by Republican presidents have ended up at various places on the spectrum between left-wing and center-right, such as Earl Warren, William Brennan, Harry Blackmun, Lewis Powell, Sandra Day O'Connor, Anthony Kennedy and David Souter. But what about Democratic nominees?
Democrats Franklin Roosevelt and Harry Truman totally remade the Court during their combined 20 years in the White House. But, since then, Republicans have appointed 17 justices, to only seven for the Democrats.
Justices such as Ruth Bader Ginsburg and Stephen Breyer, the two who were appointed by Bill Clinton, have been reliably liberal on issues such as affirmative action, LGBT rights, and the status of enemy combatants. However, there are legal theories that are more radical than the positions taken by Ginsburg and Breyer and, presumably, those are the ideas backed by Senator Harkin and likeminded colleagues.
As usual, I believe that the truth lies somewhere between these polarized viewpoints.
On the strategic question regarding the upcoming nominee, the Democrats appear to believe that the Republicans cannot hold all 41 of their senators together to sustain a filibuster against Obama's choice. Unless that person is particularly a lightning rod (a sort of leftist version of Robert Bork), they're probably correct.