This was a tough one. First, usually any bank statement that can be veiwed online has the ability to be exported also in Excel format. If this is the case, then export to Excel and highlight away.
If this is not the case or you don't have Excel, then there is a kludge. In Internet Explorer 6.X go to the File pull-down menu, select Edit with Microsoft Word. Then while in Word you can highlight the text you want. The highlighting tool is on the Formatting Toolbar. If you do not see it, you'll have to right-click on the toolbar and choose Customize. Then choose Format in the left-hand Categories box, then on the right scroll down and choose the Highlight tool. Depending on what the source web page looks like... will determine how elegant or ugly the end product is.
Not the best way, but the only free way of know of to date.