Not much.

dang explains most of the privileges / thresholds in various comments over the years:

Karma > 30 enables flagging and vouching privileges on posts and comments. <https://news.ycombinator.com/item?id=14697607> & <https://news.ycombinator.com/item?id=24134548>.

Karma > 500 enables downvotes on comments. <https://news.ycombinator.com/item?id=10224653> (Posts cannot be downvoted.)

Sufficiently negative karma causes an account's comments to be autokilled. Threshold not specified. <https://news.ycombinator.com/item?id=16409493> (Profiles can also be banned specifically by moderators, e.g., <https://news.ycombinator.com/item?id=35426426>.)

The 100 profiles with the highest karma are listed on the leaders page: <https://news.ycombinator.com/leaders>. Currently the 100th-ranked profile has 40,270 karma. There's no special privilege other than being listed, no member lounge or secret handshake.

(You can figure out most of this by searching for dang's comments with appropriate keywords, e.g., "by:dang karma threshold", which turns up most of these. I'd also looked for "vouch", "downvote", and "flag" specifically. <https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...>)