I posted not long ago about why I take street portraits sometimes and how they can lead to some great connections. There's nothing complicated about asking someone if you can make a portrait of them, but from experience there are a few things that help.
1. Having a Reason
Why do you want to take street portraits? Is it part of a project? Having a reason helps not only with your own motivation, but also to explain to your subject why you want to take their picture.
Carrying business cards is a good way of looking legitimate, as well as giving your subject a way to contact you if they want a copy of the portrait later. I also like to have a few examples of other street portraits I've taken, to give them an idea of how they might look.
You might choose to set yourself a target, for example of approaching 20 people, regardless of whether they agree to be photographed.
Decide how you want the portraits to look. Using one lens and aperture will give some consistency and you might choose to look for very simple backgrounds, or to include more features of the environment.
Busy but not too busy is my preference. Shops, waterfronts and parks all work well because there tend to be people with some time to spare.
Choosing who to ask can make a big difference on whether you're likely to have a positive response or not. Personally I tend to ignore anyone wearing earphones, with a small child, or obviously in a hurry.
As much as possible, try not to give yourself a choice once you see a potential portrait subject. Your brain will want to make excuses for not approaching, but remember that they can always just say no.
I always just try to be friendly, smile, explain what I'm doing and why I'd like to make their portrait. If there's the opportunity to chat about something other than photography, even better. Take an interest in who they are.
Using the terminology of 'make' rather than 'take a portrait' might seem unusual but the implication is much more positive and hopefully more accurate too assuming you're happy to share the image with them. Offer them the opportunity to be part of your project. Be ready to accept gracefully if they decline and be ready to shoot if they agree. They will probably be expecting some direction from you on what to do.
I don't mean persistence to keep asking if someone has already declined, but persistence to take enough time with them to get the photo you want. You'll probably be able to judge how keen the person is to be involved and how long they might want to spend with you. Our natural tendency is probably to be too polite, but remember they've already agreed to be photographed so they might well be glad that you took a little more time and care to create the best image you can.
For most of us it rarely feels natural or comfortable to approach strangers, especially in some cultures. Our brains conjure up countless reasons why it's a bad idea. A bit of thought and technique not only gives your brain less excuses but also makes it more likely the interactions will be a success.