(What actually helped us when we introduced our dog to our baby.)
You know your dog. That's the actual starting point here — not some generic advice about how dogs "should" be with babies, but the real baseline training and personality of your actual pup.
Before Baby: Get the Foundations Right
It really helps if your dog is some level of baseline trained. If they're not, put in some time before baby to do some basic treat-led training (assuming they're food motivated). For us, the single most helpful command was "house" — aka "go to your crate." It let us de-escalate any situation.
Frankly Mazie was incredibly well trained. That all fell away as her training-upkeep became priority #827. But she always obeyed "house" without question. We kept a jar of special "house treats" on top of the crate and gave her a treat 97% of the times she went to house. This obviously only works if your dog is some degree of crate trained. I wouldn't tackle that in 3rd trimester.
Plan for the Hospital Days
Make a plan for your pup. For us, we boarded Mazie both times. She got boarded when we went on vacation, so she was used to it. (If you're in LA I highly recommend Wag Hotels). We had someone on deck to take her in on the day I went to the hospital, and then we picked her up 2-3 days after baby came home.
Really it's just about only having one new creature to worry about. I wasn't concerned Mazie would be a problem. But I thought she'd be happier not getting yelled at and ignored by frazzled parents as her intro to new baby.
Lower Your Expectations
Don't expect pup to fall head over heels in love with baby. I think I hoped Mazie would become Cleo's new best friend and follow her around and babysit. Mazie was mostly indifferent. And never really played with either kid.
But she clearly thought of them as her pack. And would whine when they left the house without her. She was a herding dog — most comfortable when everyone she thought she was responsible for was in the same room.
The Actual Introduction
For actual introductions with Cleo, we had someone hold Mazie on a leash with prong collar (she was already prong trained — it was her "working" collar — DO NOT just introduce a prong to meet baby or anyone else. It's a great tool but it requires education). That person walked up to me on the couch with Cleo. I had Cleo's feet bare and let Mazie sniff and lick as long as she wanted. She did, and then eventually she got bored and wandered away and that was that. We probably did the same thing with Mouse but honestly I don't remember.
Setting Your Rules
You'll figure out your rules. Ours were that Mazie could lick baby feet til the cows come home. We didn't let her lick hands or face. Or, we tried not to. She of course got in there every now and then and I frankly didn't worry about it at all. But I'd stop her or correct her if she was licking hands and face. I didn't make her stop if it was feet.
The Barking Problem
Barking was a problem. At least for us when the kids were like 3-5 months. Little babies often sleep through literally anything. After they're less of a potato, they startle easier. When Mouse was 3.5 months, every time Mazie barked in like a "danger danger" way he'd cry. Same thing happened with Cleo.
Our approach was to (a) obviously limit barking to the best of our ability — like we put up a "do not ring bell" sign, and (b) treat it lightly in front of the kids: "Oooop! There's your puppy being silly again! Silly Mazie!" The result was that Cleo would nap through Mazie completely losing her shit. And if Mazie barked even if it startled Cleo, she didn't get scared. She didn't associate Mazie barking with trouble. And she'd heard it so regularly that it didn't bug her subconscious when sleeping.
The Reality Check
At the end of the day, your dog is a dog. Don't expect more of them than they can give. Did I wish Mazie would figure out that if the baby was sitting next to her she shouldn't bark at the door? Yes. Was that reasonable or fair of me? No.
A Word on Grandparents (And Setting Boundaries)
Finally — a word on grandparents. I don't know about you but "dogs in the house like people" is weird to some of the grandparents in our life. Ignore them. You set the rules. You control your relationship with your dog and between your dog and your kid.
You know your animals. As long as you don't expect more from them than they're capable of, they're probably the most predictable creature in your house. So when you're teaching the grandparents how to grandparent (yes, this is a thing and it sucks), include animals in that education. "Here's what we let pup do. Cat is/isn't allowed in nursery."
I find it helpful to differentiate between being aware and worrying. "I'm not worried about it" was my key phrase with Mazie. I was aware that she licked his toes. I wasn't worried about it. I was aware that she licked Cleo's hand during dinner and Cleo kept eating. I wasn't worried about it.
Part of our Newborn Survival Guide.