When you walk around you leave your genetic information everywhere. Right now it's not worth it for anyone to do anything with it, outside of DNA profiling in forensics, but sequencing keeps getting cheaper so this may change. Long term, I don't trust genetic data to stay private.

For the near future, however, the details of your genetics probably will stay private. If you want them to. You could, however, sign up for something like the Personal Genome Project or Genomes Unzipped which would make your genetic data public, along with other medical details. If you look at this example consent form there's a long section on the risks, for example:

7.3 ... if you and your partner are both participants you may learn that both of you are carriers for recessive disease-causing variants in the same gene, suggesting a higher risk of severe disease in your children.

This is not what I would consider a risk, in that you still have the option to ignore the information and do what you would have done anyway, you just might not want to. Looking over their other listed risks, the only one that seems significant to me is:

7.1.c Whether or not it is lawful to do so, you and/or a member of your family could be subject to actual or attempted employment, insurance, financial or other forms of discrimination or negative treatment on the basis of the public disclosure of your genetic and trait information by GNZ or by a third party. Although some countries have laws that prohibit certain forms of genetic discrimination, these laws may not apply to you, may not protect against all forms of discrimination or may not stop a third party from discriminating against you even where it is prohibited by law.

The big issue here is health insurance. Economically it's run kind of like an independent insurance business but because what so many people actually want is health care for everyone with shared costs we've tangled it up in knots. So while you might expect insurance against health risks to cost in proportion to those risks, in the US we have the Genetic Information Nondiscrimination Act (GINA) which prohibits insurance companies from denying coverage or charging different prices based on genetic data.

Another issue is employment. If it's public information that you're at high risk for an expensive disease, employers might not want to hire you because we run health insurance through employers. Employers also might not want to train you if they knew you were likely to get a debilitating condition. (Perhaps an astronaut-training program would want to screen out people with heart problems) For both of these, however, GINA prohibits them from doing this.

GINA doesn't cover other kinds of insurance, like life insurance, however. Which makes some sense:

Michael Snyder, the scientist from Stanford, got a glimpse of that himself. After he was sequenced and found out he was at high risk for diabetes, his wife tried to increase his life insurance. (npr)

If people who get unusually worrying results from their tests go to buy life insurance then you have adverse selection, and life insurance gets more expensive for everyone while transfering money to people who got bad news on their tests. One way to deal with this is to buy life insurance before being tested, another is to just decide you don't need it, but if you want life insurance but don't want to buy it yet it's a significant concern.

Are there other kinds of genetic discrimination or other reasons not to make genetic data public?