It’s a seductive Beltway storyline: The President is in danger of losing liberals by ceding too much ground to Republicans. That’s the subject of a big Washington Post story today, and there will be plenty more along these lines.

In fact, despite the loud criticism of Obama from prominent lefties, liberal and Democratic rank and file support for Obama remains solid. The one who really has the most to fear from an angry base is House Speaker John Boehner.

The Post reports:

Key liberal groups, which helped elect Obama in 2008, are raising concerns that he has given up political ground to Republicans, allowing the message of reducing government to trump that of creating jobs and lowering the unemployment rate.

Seizing on Friday’s deal, which would cut $38.5 billion from the fiscal 2011 budget, activists on Tuesday threatened to sit out the 2012 presidential campaign if Obama goes too far with further cuts.

While it’s a bad idea to cut spending before the economy has completely recovered, the details of the budget compromise that have emerged suggest that Obama may have gotten a better deal than expected.

And while there’s a lot of disappointment among liberal elites and activists about Obama, the truth is that he hasn’t lost much standing with rank and file liberals or Democrats.

Gallup’s weekly demographics poll shows Obama’s approval rating among liberals and Democrats has been relatively stable over the past month. A recent CNN poll also showed that Democrats and independents broadly approved of the budget compromise even before the details were really out, which makes sense since unlike Republicans who seemed eager for a shutdown, Democrats tend to like compromise.

Indeed, it’s precisely because Obama’s standing among liberals and Democrats is so strong that liberal activists and elites have to make so much noise to hold his feet to the fire. Conservative elites, through an incredibly influential media ecosystem that includes Fox News, Rush Limbaugh and others, have much more influence over the opinions of the conservative base than liberal elites do over theirs.

Boehner is the person who really has to worry about pleasing his base. That same CNN poll, while giving him broad approval ratings among Republicans, still showed that a bare majority of GOPers believe he has given up too much ground, and his approval ratings among conservatives and Republicans are far lower than Obama’s standing among liberals and Democrats.

This is partly why Obama has more leeway that Republican leaders to compromise — and why it’s all the more important for liberal groups to pressure him to prevent him from giving too much ground.