Generally, "walk a fine line" means to handle a situation with great delicacy and care, with not a lot of room to maneuver (figuratively speaking). Usually the stakes are considered to be high. Often it's understood as trying to balance two extremes. (Of course, a specific context might be understood as a variation on this...)

For example:

"Rosa walks a fine line in renogotiating higher prices with our best client."
"The police chief is walking a fine line in dealing with the kidnappers."
"Politicians walk a fine line between appealing to the most voters and watering down their messages."

It's based, I'm sure, in a tightrope walker (funámbulo), who walks an extremely fine line and the risks are great if he fails, so...

...in Spanish, "andar (o bailar) en la cuerda floja"... is used this way, I believe.