If you are using React Router, I'd suggest looking into the react-router-bootstrap library which has a handy component LinkContainer. This component prevents default page reload so you don't have to deal with the event.
In your case it could look something like:
import { LinkContainer } from 'react-router-bootstrap';
<LinkContainer to={givePathHere}>
<span className="upvotes" onClick={this.upvote}>upvote</span>
</LinkContainer>