Thankfully Google has the answer to that problem. The company has recently released a wired adapter for the Chromecast. Basically this is a little brick with an Ethernet port so that you will be able to directly plug your modem into the adapter, and the adapter will then connect to your Chromecast via a cable, thus ensuring that it will have a decent and stable connection for you to stream your content on.
It is a pretty simple setup but it seems to be good enough to get the job done. It is priced at $15 which we suppose is a decent price to pay, although the bad news is that either Google did not stock up on enough units or the demand was too great because it looks like the adapter is currently out of stock. We expect that Google will eventually make it available again, so you might want to bookmark its product page and check back later for availability.