The Left Bank is a decision available to 
France. It is available if 
Prussia exists and owns the to provinces Trier (In Rheinland) and Cologne (In Nordrhein). Prussia owns both provinces in the beginning of the game. Otherwise it requires that France has researched Ideological Thought. It cannot be taken if Prussia has formed 
North German Federation or 
Germany, so it must be used early in the game.
It reduces the relation with all states in Germany by 100, gives 4 infamy and 11 prestige and gives an acquire state casus belli to France against Prussia. The casus belli only lasts for 1 month, so war has to be declared immediately.
It is part of the chain of decisions surrounding the Rhine crisis. It leads to the Prussian decisions Die Wacht am Rhein and Niederwald Denkmal. The two decision will remove 7 infamy from Germany, but will not affect France directly, so one should not avoid taking the decision in fear of those two decisions.
The AI will always take the decision, unless infamy is below 0.5. Taking the decision, doesn't mean that France will then naturally always use the casus belli though.