What are some good ways to approach professional people that I don't know on LinkedIn?

As ShaRon had mentioned, I don't accept LinkedIn invitations from people that I don't know. Part of the problem is that most people use the default message that LinkedIn provides you when you'd like to connect with someone. It goes something like: "Hi Dave, I'd like to join your LinkedIn network". Really? No Kiddin!
If you don't know your connection, use that connection message to add something more personal, like " I saw your blog post on CareerVillage and we seem to have mutual interests in the area of blah, blah, blah..." At least it will give the person you want to connect with a reason to consider why you might be an interesting person to network with.
I hope that helps!

Be honest on while you want to connect with them. Career interest? Education purposes? Mentor? Ensure that your profile is complete before you start connecting.
Take time to view teaching videos on how to effectively use LinkedIn.
YouTube has an extensive library you can explore. Here is an example: https://www.youtube.com/watch?v=_74i-CJmRCI

I personally do not accept invitations from strangers on Linked In. If I do not know the individual and have not worked with them in the past I ignore those request, so I understand why you would pose this question. I would encourage you to watch tutorials on youtube to get a better understandong on how to use LinkedIn as a networking tool. When you reach out to strangers, share that you are a looking for a connect in the industry to assist you with getting your foot in the door. I would apprecaite the honesty instead of getting random request to join my network by folks I think are just trying to use me for their personal benefit.

Also I am old school so I would be more inclined to assist if I got a personal note not the generic "I'd like to connect with you." Good Luck

The best way to approach people you don't know on LinkedIn is to be professional and courteous, but also brief. If you are looking for advice/input, say so - but remember, sometimes people don't go on LinkedIn every day or they may not reply to connection requests/InMails at all so don't be discouraged.

Another way to connect with people you don't know on LinkedIn is to look for a common thread on their profile and how it relates to your background, such as organizations you may have worked, industry associations you might share, similar fields of study, or if you went to the same high school or college. That gives you an "in" or an angle on how to approach them and ups the likelihood that they will respond to you.

One example of how I did this was when I was trying to track down a vendor at a company I had never worked with before. As luck would have it, I searched on LinkedIn for people who worked there that went to my college and there were a few folks on that list. I then proceeded to reach out to them, calling out that we were both graduates and mentioned that I wanted to learn more about their company and offerings. It worked and I was able to not only make a connection that has served me long into my career, but I got the information I needed as well for my higher-ups.

One of my favorite syndicated career columnists, Liz Ryan (I view her as the Suze Orman of career advice), wrote an article in Forbes that sums up how to reach out to people on LinkedIn:

https://www.forbes.com/sites/lizryan/2017/03/01/five-ways-to-be-polite-on-linkedin/43544a329516

The Muse (a career advice/development web site) also has a great article on Do's & Don'ts for connecting with people on LinkedIn:

https://www.themuse.com/advice/24-linkedin-rules-you-might-be-breaking

I am forwarding a link that will help very helpful in forming and maintaining networking relationships. Based on my many years in Human Resources, I deem this to be very important to read and understand and apply, as I used this concept in a program which I created to help laid off workers locate jobs by networking.

https://www.themuse.com/advice/why-arent-i-hearing-back-about-jobs-im-applying-for?bsft_eid=ce69fe67-a25b-6f0f-46a7-3ca16b93938e&utm_campaign=daily_20180604_full_post_22746&utm_source=blueshift&utm_medium=email&utm_content=daily_20180604_full_post_22746&bsft_clkid=d7ab3c79-d530-4bc5-b220-c198c70827ca&bsft_uid=54658fa1-0090-41fd-b88c-20a86c513a6c&bsft_mid=e099f609-6470-4917-8466-da23de939177

Hi. I do accept requests on LinkedIn from people I don't know, but I always study their profiles and check out who else they're linked to find people I know.

I'm also much more likely to really connect to people who pose questions about networking, my profession, my alma mater, etc. Those are the people who seem serious about using LinkedIn to make lasting connections that will help them professionally. I'm less interested in those people than members who simply are trying to rack up connections.

Hi Hollie!

You asked a very important question!

Here is some of the best advice that I have seen on networking:
http://www.wikihow.com/Network
https://www.themuse.com/advice/nonawkward-ways-to-start-and-end-networking-conversations

Remember, the best way to get a job is to ask for information. When people know that you are sincerely seeking information, many interesting things may happen as information is exchanged and relationships begin to develop

Please let me know if this is helpful. I would like to follow your progress. Please keep me informed. Best of luck!.