If using Chrome
https://support.google.com/chrome/answer/7623121?hl=en-GB is probably a good idea. Comes with a hit on resources, naturally.
The patch will, however, prevent rogue Spectre-abusing scripts running on a browser and plundering the memory of other sites - it isolates every site and, indeed, frame, within its own process and the Spectre exploits as far as I'm aware cannot jump out of their own process context.
Meltdown received the coverage because it is, by a mile, worse in terms of impact and easier to exploit. Spectre can be mitigated to some extent through recompilation of binaries and ensuring appropriate isolation of workloads to processes.
EDIT: Microsoft have recently implemented something similar in MS Edge and IE 11. Grab those patches.