What was the problem at Babel? In the Genesis account of the tower of Babel a bunch of guys got together to build a really tall tower somewhere in the desert. God gets wind of the affair but it doesn't sit well:


  Genesis 11:5–6 (ESV)
  And the Lord came down to see the city and the tower, which the children of man had built. And the Lord said, “Behold, they are one people, and they have all one language, and this is only the beginning of what they will do. And nothing that they propose to do will now be impossible for them.


So what does He do? He scrambles all their language as a punishment and their plans are foiled.

What was the problem with building a tall tower? Are modern skyscrapers a problem for God? Should the guys in the Burj Khalifa be worried?
The problem was not with the building, but with the intent of the builders, as shown in verse 4:


  They said, "Come, let us build for ourselves a city, and a tower whose top will reach into heaven, and let us make for ourselves a name, otherwise we will be scattered abroad over the face of the whole earth."


This is in direct opposition to what God had just commanded Noah a few years before (Genesis  9:7):


  As for you, be fruitful and multiply; Populate the earth abundantly and multiply in it.


The people in their journeyings found a place they liked, and decided to quit following God, and rather settle where they were comfortable. On top of this, they displayed the hubris to build a tower to "reach heaven" - ie, to find God on their own, rather than through His intended means.

God's judgement was not upon the tower, per se, but on the people. Nowhere else in scripture can I think of a place where towers are forbidden or condemned (indeed, almost all ancient city states had towers for protection - and Jesus references a tower that collapsed onto some people in Siloam in a neutral fashion).

The "problem at Babel" was pride, arrogance, and an unwillingness to follow God's command to replenish the earth after the flood.