If you can get a visa to come to the United States, such as a tourist visa, you can apply for asylum when you arrive, by telling the inspections officer that you fear returning to your country and wish to apply for asylum. However, it's best to wait, and not to mention this unless the officer is acting like he or she doesn't plan to let you into the U.S. anyway. If you start the application process now, you'll have very little time in which to find a lawyer or prepare.
If the inspections officer allows you in to the U.S., then you can apply for asylum by mailing in an application, with supporting documentation, on Form I-589. The form and instructions are available for free download on the USCIS website. However, if at all possible, you should get help from an attorney.
There are many reasons to avoid requesting asylum at the port of entry to the United States.
One is that the inspections officers have the power to quickly find you inadmissible and deport you, in which case you will not be allowed to return for five years. This can happen if an inspector believes that you are making a misrepresentation (committing fraud), or misrepresented the truth when you got your visa, or if you do not have the proper travel or visa documents at the time you request entry.
This quick deportation procedure is known as "summary exclusion." It can be applied to anyone except people entering the United States under the Visa Waiver Program (according to a 1999 decision by the Board of Immigration Appeals).
There is an exception to the summary exclusion process for people who fear persecution and request asylum. So, even if you do not have the proper documents or you have made a misrepresentation, you could still be allowed to enter the U.S. if you make clear that your reason is to apply for asylum and you can show that you'd be likely to win asylum.
Another reason is that the Trump Administration has begun treating asylum seekers as unlawful immigrants, and trying out a variety of policies to deter them, such as placing all of them into detention, charging application fees, or forcing them to wait in Mexico. We won't detail these here because many are the subject of ongoing litigation and might have changed by the time you read this.
After you have said you want to apply for asylum, you're supposed to be immediately given a "credible fear" interview by an asylum officer. The purpose of this interview is to make sure you have a significant possibility of winning your case. Most importantly, the officer will want to be sure that your request is based on a fear of persecution. This interview is supposed to be scheduled quickly, within one or two days, but it has been taking longer.
If the officer isn't convinced of your fear, you must request a hearing before an immigration judge. If you don't, you will be deported from the U.S., and not be allowed to return for five years. The judge must hold the hearing within seven days, either in person or by telephone.
If the judge finds that you have a credible fear of persecution, you'll be scheduled for a full hearing. In that case, you should seek an attorney. This proceeding will take place in Immigration Court, before a judge, and with an attorney representing the Department of Homeland Security (DHS).
Most asylum applicants are held in a detention facility at this point. Depending on the latest policy it might be possible to request release, called "parole."
The applicants most likely to be granted parole are those who can verify your identity, have family or other contacts in the area, can post a bond (money that's given up if you don't show up for future hearings), and can show you'll be financially supported until a decision is made on your asylum case.
If you fail to convince the immigration judge of your credible fear of persecution, you will be deported (removed) from the United States.