As a designer and manager who has actively recruited designers for over 10 years, I get asked to connect people with design talent often.
Starting on Linkedin might seem a bit spammy to designers you don't know. People's time is valuable, so I wouldn't expect most seasoned designers to jump up to meet you for coffee unless you have a pre existing relationship – or your idea is really *that good.*
How do you meet designers then? Ask your IRL network – coworkers, friends, family. Most people know a designer and you'll get referrals, but your job is to find someone who's a fit for you. You will want to filter out designers in the broader sense: keep to product and UX designers (as I'm assuming your product is mostly digital.) You can also post on job boards like Dribbble, Angelist, Unicorn Hunt, etc, but start with people you know and are close to you.
Generally, great designers are compelled to work on projects that will allow them to design solutions that are their best work yet. Start out by getting them hooked on your idea and then start to shape out how you might work together.
To respond to your question of senior vs junior, it would help to know your motivation: is this monetary/equity based? I would forget about level at this point and find someone who:
1. Has the nature of the work you're looking for in their portfolio or a strong conviction to create like work
2. Feels like a cultural fit with you
3. Believes in the concept itself
I'm happy to follow up with a call if you'd like to talk more. It would help a ton to know more about you. Are you a developer? Idea person? Marketer? I can frame out an ideal way to find your cofounder with a bit more information.