Both include a Stabilizer effect.
How you apply it depends on which program -- and which version of the program-- you're using.
Welcome to the forum.
As Steve points out, both flavors of Premiere include a form of stabilization, and Adobe After Effects has one, that is even better.
ProDAD's Mercalli is a stabilization plug-in (not sure if it works with both PrE, and PrPro, but think that it does), and many sing its praises: http://www.prodad.com/home/products/video_optimization/300391667,l-us.xhtml
The plug-in is quite expensive, more than I could justify. But it is bundled with some competitor products (e.g the top version of Corel VideoStudio). VideoStudio regularly pops up on Amazon lightening deals and I bought a copy just for Mercalli and use it as just another preprocessing tool to produce stabilized clips for PRE.
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That is a great way to do it.
I have also seen several ProDAD products go on sale at Digital Juice. Some time back, I picked up Heroglyph, VitaScene and Adorage for about US $ 120. Were I looking for Mercalli, I would poke my head into Digital Juice about every month, just in case.
The stabilizer included in PRE11 is in fact the New Blue filter, which in it's own right is a superb filter, unfortunately there are limitations in the PRE11 API which prevents the use of certain functions of the effect, making the stabilizer virtually useless in PRE except for very minor shake and installing the stand alone New Blue effect makes no difference at all, the limitations are still there. Trying to get this rectified has been a continuing saga between myself, New Blue and Adobe.
I am really hoping that this will be corrected in PRE12, but in the meantime I use Corel VideoStudio X5 Ultimate to correct stabilization using either the included ProDAD version or the fully functioning version of the New Blue plugin. As Bill said, the stabilizer included in AE is very good, although even in AE you have no real control of the effect, but it does work, so why is the PRE stabilizer less functional.