Erskine's original Super Soldier Serum does two main things - it gives the user peak human physical capability and enhances their existing traits - or as he put it, it makes you "more".

Bruce is already at near-peak through his own strenuous training regimen so we might see a minimal improvement to his physical prowess. If that was all it did we would likely hardly see any change to the status-quo. However it is the mental changes of the semen that would have the greatest effect on Bruce, while it would enhance his existing detective skills it would also significantly amplify his paranoia, aggression and guilt which is what was originally drove him to become the Dark Knight in the first place.

Having all these aspects of his persona focused and amplified would create a much more terrifying version of the Batman - one who is far more colder and calculating and proactive when it came to dealing with the criminal element. This is a version of Bruce who may have no qualms in killing his enemies even before they become a threat in the most pragmatic way possible. Even his friends would be at risk, we saw the kinds of secret contingencies that he developed to use against his teammates except through the amplification of his distrust and paranoia he may see fit to use these plans proactively and without restraint - all for his own interpretation of the greater good and self-styled "justice". All in all, a juiced-up Batman is not something anybody would want to see because I could easily see him changing over time from the paragon of justice into something similar to the Punisher in his ideals and approach to things. This is a Batman who doesn't just make contingencies for any situation he can conceive of, this is one who also proactively implements them whether the individuals in question have already proven the need for it. Essentially adopting a mantra of "Guilty until proven innocent" and acting accordingly.