I don't think there is an SLDS class specifically that enables scrolling, however an easy way to enable this using standard CSS is to do the following:
<style>
.scrollable {
height: 400px; <!-- Don't need this if it's okay for height to be dynamic -->
overflow-y:auto;
}
</style>
<div class="slds-page-header">
<p class="slds-text-heading--label">Contacts</p>
<h1>My Contacts</h1>
</div>
<div class="scrollable">
<!-- The rest of the page goes here -->
</div>
EDIT: As @techbusinessman wrote in a comment below, there is now a standard utility class for making a div scrollable. The documentation can be found here: https://www.lightningdesignsystem.com/components/utilities/scrollable/#overview
The code would now be as follows:
<div class="slds-page-header">
<p class="slds-text-heading--label">Contacts</p>
<h1>My Contacts</h1>
</div>
<div class="slds-scrollable">
<!-- The rest of the page goes here -->
</div>