The Big Banks of the world, such as Barclay's, Lloyd's, Santander & many others, have had far too much power for far too long, Breaking them up would mean giving more power to the people and give them a chance to regulate and keep safe their own hard earned money. Which would lead to the banks making less of a profit meaning they have less control over the people, which they shouldn't have in the first place.
It's common knowledge that diversity is key to a strong economy, that's why many of the developed worlds' governments have controls in place to prevent monopolies of certain industries. If the big banks were broken up it would encourage competition and maybe we would see economic growth as a result.
The monopoly that big banks encompass encourages large corporations and the wealthy to continue to increase their wealth at the expense of individual citizens and government. Big banks are provided with tax incentives, tax breaks and other tax payer funded bailouts that smaller banks are not. In addition, smaller banks generally keep local monies local - supporting the community members' own neighbors, whereas large banks' monies often leave the community.
No, I do not believe the big banks should be broken up. The larger a company is, the more buying power it has, and the cheaper it can provide its services. I believe that the big banks can provide loans and mortgages at lower interest rates because of the amount of volume they deal with benefiting people using these services.