We have a huge amount of backlinks from good sources, but unfortunately a large chunk of them are nofollow.
I'm just hoping these aren't completely useless for SEO.
Update:
I edited the question to make it clearer that we have thousands of incoming no-follow backlinks.
There are many different opinions, on this question, but the most important thing you need to know about it is that it won't harm your seo if you use nofollow for external resources, also it won't harm your SEO if you don't use them.
This is how page ranking works, the most external websites linked to your site you have, the better page ranking would be served, accept links to external pages as "votes" to make it cleaner. If you want to "vote" for that websites you can leave the anchor and let the robots follow it, but if you don't feel like voting for that page you can use nofollow attribute, this way the external link won't get your "vote" because the robot won't follow it, also you are not getting penalize for using nofollow attribute, no matter on how many pages and links you have it.
Use it when you link to paid links and untrusted one, don't use it if you actually want to give that link a "vote".
It is really bad if you have internal links pointing to your website pages and they have attribute nofollow, unless you want to use it for pages like register/login/search modules, you can nofollow them but not the content you want to index.
It is good if you have attribute nofollow only on external links you don't want to "vote" for, and as mentioned above on some pages you want to nofollow because they aren't usefull for the ranking.
The more pages linked to your website, the better page ranking would be served, unless they are spam websites, or untrusted one.
P.S. I never made it to say thanks to dan because his encouraging comments, so thank you dan, you really make stackexchange better place for me.
With regards to the main search engines (eg. Google), incoming/inbound nofollow links are not going to influence your search engine ranking directly (for better or worse). At least, they shouldn't.
However, as with any inbound link, it can drive traffic to your site. If these visitors then share/link your content as a follow then obviously this secondary action can potentially influence your ranking. So, indirectly, these nofollow links can influence your search engine ranking / SEO.
