If I am grappling someone while I have the Blink spell activated, does that person travel with me or is the grapple broken?
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From the Grappled condition in Appendix A of the Player's Handbook or Player's Basic Rules:
So if you use Blink and vanish into the Ethereal Plane, the grapple will end. Obviously if you use Blink and don't vanish into the Ethereal Plane, the grapple will keep going until you do.
Yes it breaks grapples, no you can not take somebody with you.
Pertinent part of the Blink spell that pertains to your question:
The spell doesn't say you can take somebody with you. The spell is also specific in that it states you move between the planes. It does not indicate an area, or multiple creatures.
Now, if the target you were grappling was able to interact with the etheral plane, or if you were able to interact with the material plane from the ethereal plane, then you could maintain the grapple.