Easy Board Games

Candyland, and Chutes and Ladders are classic starter board games.
It's tough with a baby around because everything is a choking hazard till they are about 3 and even then they'll shove beads/marbles/crayons in their ears and up their nose.
You need safe zones for the baby where older kid things are not within their reach.
Playpens used to be great for this.

I think there are very few board games without small pieces. The best thing to do is to keep the board games in the 6 year old's room or somewhere out of the baby's reach and have your 6 year old play them when the baby is not around (during the baby's naptime for instance). That's what I do.

One game I can think of without small pieces is Guess Who? It's a fun game for a 6 year old.

snakes and ladders and candyland are very easy, but might be too boring for a 6 year old. by this age he's probably ready to start monopoly and risk, but of course they have small pieces.
i'd play the games that challenge the older and are more fun for him, and just be super vigilant about the pieces.
or start card games.
khairete
S.

Engineering Ants is a fun cooperative game. You work as a team to build contraptions to get around hazards before the anteater gets to the anthill.

Pie in the Face is the new favorite at our house. (Although the cleanup isn't fun.)

We play UNO and Skipbo a lot. Old Maid and Crazy Eights get pulled out once in awhile. Headbands is fun.

My kids learned Checkers around that age. They also liked Trouble. Twister would be fun.

Other than card games it is hard to find games that don't have dice or playing pieces. You just need to keep the games stored up high, play at a high table, and clean up pieces.

Dominoes with large pieces. Different colors for different numbers can help so they aren't always counting or memorizing the patterns.

Uno - if he can't hold the cards, get one of those doodads made for that purpose. They are made for kids and also for older folks with arthritis.

Get a book of card game rules, and use regular cards for dozens of different games. Get extra decks because someone always loses a card at some point!

Yahtzee Jr, checkers, card games, Jenga, Sorry.

I agree with the others who suggested playing board games with your 6 yo when the baby is napping, contained in some way or being watched by someone else.

Updated

Yahtzee Jr, checkers, card games, Jenga, Sorry.

I agree with the others who suggested playing board games with your 6 yo when the baby is napping, contained in some way or being watched by someone else.

Candy land - put away the regular plastic pieces and use something safe for the player pieces. You could do the same for chutes and ladders.

My 6 year old likes card games like go fish or rummy. He also likes Uno.

Jenga

And it's not a board game but in the vein of things to sit and do together, my 6 year old likes to do puzzles and you can get them with bigger pieces.

The Spot It card game, Slamwich, UNO (previously mentioned) are all good ones.

I think a different question would be 'how do I contain game time/baby so that we can play with pieces". I would suggest a pack and play with toys near the game table, or putting a high chair with something for baby to play with on the tray; even an umbrella stroller and a basket of toys baby likes to have on hand to pass to them. Do a quick survey of the area when you are done. Another idea is to divide and conquer; set up a gate in your son's room and a card table or play on the floor while your spouse is home; one of you takes baby, the other plays with your son.

This one's pretty easy and fun: https://www.amazon.com/Peaceable-Kingdom-Treasure-Winning...

It has dice though which is still a choking hazard. You'll just have to store it away from baby and play when baby's napping or otherwise occupied.

It's really hard to keep the toys separated, I know because I have a lot of years in child care.

I can't think of many games you could do but I will suggest that you have a space that only the older one can go in so they can have their own toys that are not for infants. Put a gate over their door so that the baby can't get in and then the older one can have fun without it risking the little one.

When you get out something as a family that has small pieces put the little one in their high chair so they're not on the floor. It makes them table height too so they're not left out.

Hedbanz (This is a family favorite)
Monopoly Junior (some small parts)
UNO
Twister

You could also teach your child games with regular playing cards.

We had candy land which is a lot like snakes and ladders. Easy for kids to do and they find it really fun. We got Jenga after our kid's teacher brought it in to class. It's a tower of blocks stacked in threes, and you pull one block out at a time, trying not to let it topple over. Our older kids will play too if we pull it out. There are child versions of Monopoly. We had one. We just didn't follow all the rules (we made it basic) and our kids loved that. We have apples to apples for kids (just got the adult version) and the kids have enjoyed that. Good for learning to read too. We have Trouble - at 6 they can do that (where you move your markers around the board by popping the dice in the middle). We do a lot of Go Fish (very popular with our kids when small). We have Uno but haven't played it yet - heard that was good too.
A popular one we had for 2 players was Guess who it is or Who is it? where you flip over all the people wearing hats .. then next time you might ask "Does your person have a beard?" and they flip over all the ones who do not have a beard. I liked that one - goes quickly so even if you have more than 2 people, you can play the winner, etc.

Pie Face is my son's favorite, but we don't play as often as he'd like because of the mess. Our dog loves it also, because of the mess!

We gave our son the 5 Second Rule for Christmas, and he loves it. He also likes Shelby's Snack Shack, Hoot Owl Hoot, and Jenga.

Maybe a card game like Memory?

All I can think of that my daughter LOVED at age 6 (and still loves - she recently turned 7) all have small pieces. They are for older kids but her very favorite games are Monopoly, Life and Settlers of Catan. She's crazy about Settlers of Catan. Her other favorite game is Mancala. She also likes Blokus. All of these have small pieces though. Sorry. I can't think of a single game without small pieces. Cards? She likes Uno a lot. She also likes playing the card game "golf" which you can do with any pack of cards.