There are a series of issues of law and probably more important, practicality.
Apart from SEC and NASD laws and requirements -- and you can ask those organizations but USUALLY only recent securities, fraud or financially related convictions serve as a bar -- there are the standards of the firm that would hire you and the potential need to have you "bonded".
I have no idea as to whether the brokers would hire a recently convicted or paroled felon or if insurance companies that do the bonding will look askance at a drug conviction.
In my experience it often depends on the exact facts and circumstances underlying the offense, your entire prior record and history (good and bad), and what the prosecutors and police may say in terms of their assessment of your culpability and cooperation, if they say anything.
In other words, if you had earned an MBA from Harvard Business School, had a career of success in business, then had a short bad spell and did drugs and got caught, accepted responsibility, rehabilitated yourself and turned your life around, and have the support systems in place that employers know is likely to cause you to maintain yourself with a good track record, and are now clean and can always pass drug tests, they'd find a way to get you in the business. There's a guy formerly from Texas that sort of serves as a good example in that regard, G. W. Bush.