Lets break this into two parts - getting the visa, and using the visa.
Getting the visa.
No, you will not have any issue as long as your passport is valid at the time you apply. Neither the issuing of the visa, nor the time that the visa is valid for will be impacted by the expiry on your passport. Even after your passport has expired the visa will still be valid - you'll just need to present both your expired and new passports at that time. This is true for citizens for any countries.
Using the visa/entering the US.
In general, the US requires your passport to be valid for at least 6 months after the date you will LEAVE the US. Thus if you were planning to stay for one month, your passport would need to expire in at least 7 months time.
However there is a list of countries for which this rule does not apply, know as the "Six Month Club". Citizens of countries that are on this list only need their passport to be valid up until the time they plan to exit the country, not and longer. You can find the latest copy of the countries on this list here, and at this time India IS on that list, so you are good.
Note that when you enter, if your passport has less than 6 months validity remaining, you will only be granted entry to the US up until the date your passport expires, not any longer. Thus if you were to enter the US a week before your passport expires, you would only be allowed stay in the US for one week.
Having said that, if you have the time I would strongly recommend getting a new passport. It is likely you will be given a 10 year visa, which means that on any subsequent trips to the US within the next 10 years you will need to carry 2 passports - your current one plus the one with the visa. It would make more sense to get the visa in a new passport and avoid that and any other potential issues.