Try Not to Take It Personally
There should be no reason for your man's ex and his child's mother to hate you. After all, unless he cheated on her with you, there's no reason for her to spew hate your way, right? If only it were that simple, but it's not that easy. Most women who have a man's child probably thought they'd marry him. They may have even gotten pregnant purposely in order to "keep" him. If she loved him, and still does love him, she probably doesn't like you because she's a bitter ex harboring ill feelings; and because she can't have him and hasn't moved on, she focuses her resentment towards you. There's nothing you can do about that except ignore her and try not to take it personally.
You may ask your man about the relationship and how it ended, but at the end of the day, his version and her version of the relationship will always be different, so trying to understand why she doesn't like you might be futile. Just try your best to keep your distance and allow her time to get over it.
Let HIM Do the Talking
Once you realize this woman may never like you, it might be hard to keep your mouth shut if she pops off on you for no good reason. As tempting as it may be to give her the business, let your man do the talking when it comes to his ex. Arguing with her will only escalate the situation, and if she knows how to push your buttons, she can trigger a situation where she can actually cause conflict between you and your boyfriend.
Avoid potential confrontations by telling your man how you feel and then let him handle her. The last thing you both need is to be fighting over something his ex said or did, so tell him what upsets you and then see what he does. If he does nothing, then this might not be the relationship for you because your feelings are just as valid as hers. But give him a chance to be a man and handle his business, and keep your comments to yourself.
Regardless of whether or not his ex takes a liking to you, you must always carry yourself in a way that shows respect. After all, she is his child's mother and she will always play a major part in the child's life, so you have to accept that...as difficult as it may be. Unless you are explicitly invited to be a part of the child's life in a meaningful way, involving yourself in sensitive issues may only cause more drama, even if you have good intentions.
No matter how your man and his ex feel about each other, the child sees them as mommy and daddy, so they both have to get along for the child's sake, which means you also have to have a good relationship with your boyfriend and the child's mother. She's not going anywhere - no matter how much you wish she might disappear - so support him, respect her and realize that if you are to remain in this relationship, she will always be a constant.
Talk to Her...If You Can
She is probably the LAST person you want to talk to, but maybe all you need to do to clear the air is have a friendly little chat. She may feel that you're trying to take over her role and be the "fairy step-mother"...thus causing her to feel a little jealousy or resentment. But if she has a chance to meet with you and see what your intentions are as they pertain to her child, you can explain that you only have her child's best interests at heart and you are in no way trying to replace her.
Invite her to coffee and let her know that you respect her role as the child's mother and your boyfriend's co-parent and that you simply want to support them in any way you can while staying in your own lane. If she's a reasonable, sane person, this may be all that's needed to keep the drama on the back burner. If she's not, then hey..at least you tried.
Keep Your Business to Yourself
In the event that you actually become "friendly" with his child's mother, realize that she is still the ex and that anything that goes on in your current relationship is strictly off-limits. Also, if your man is civil, or even friends, with his ex, you might want to remind him that any details of your relationship should remain private. They probably speak regularly since they share a child, and since they have a history, he may feel comfortable talking to his ex about everything...especially if you two have an argument. Let him know that while you respect the bond they share as parents, that bond should stop there and any discussions with her that involve the two of you would be inappropriate - especially if she's not supportive of your relationship.
If she wants him back, she could use anything he tells her against you in order to make an argument for why they should be a family. And if she doesn't want him back, you don't want her thinking that you're a ship passing in the night and there is no stable environment for their child. All she needs to know is that you are a supportive girlfriend and you two are good. That's it.