You need to change the doctype in the following files:
app/design/frontend/<your_package>/<your_theme/page/1column.phtml
app/design/frontend/<your_package>/<your_theme/page/2columns-left.phtml
app/design/frontend/<your_package>/<your_theme/page/2columns-right.phtml
app/design/frontend/<your_package>/<your_theme/page/3columns.phtml
app/design/frontend/<your_package>/<your_theme/page/empty.phtml
app/design/frontend/<your_package>/<your_theme/page/popup.phtml
app/design/frontend/<your_package>/<your_theme/page/print.phtml
In those files, replace the following line at the top of the file:
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
With:
<!DOCTYPE html>
Regarding any downside, I'm afraid this is not Magento related so I suggest you check this very good StackOverflow post regarding reasons not to start using html 5 doctype