The First Minister said a border ban would be a "partial measure at best" and pointed to the fact that even an essential-only travel rule would see thousands of people cross the border legally – such as for thousands of people living in Shropshire and Mid Wales.
Of the 12,431 Indian variant cases so far confirmed in the UK, the vast majority are in England with 10,797 cases, while Wales has just 97.
Scotland has 1,511 cases of the variant and has imposed restrictions on travel between the council areas of Bedford, Bolton and Blackburn and Darwen, with First Minister Nicola Sturgeon telling Scots to delay plans to visit those areas.
But Mr Drakeford said there were no such plans in Wales, saying: "Even when our border was closed with England, thousands of people crossed the border every day for reasons that were allowed in the law.
"If you lived in Wales and worked in England or the other way around, you were able to travel for work purposes. If you were attending a hospital appointment you were able to cross the border. If you lived in England and your school was in Wales then you could still travel.
"So saying we would prevent travel from England into Wales is a partial measure at best.
"And secondly, when we were in a position where the whole of England was in a more difficult position than Wales, at least for policing purposes that was relatively straightforward.
"In a practical world, how would a police officer know whether somebody crossing over the border was coming from a hotspot in the north-west of England, or coming into Wales from a part of England where there was no significant problem?
"So just in a practical sense it's much more difficult to take that action at the moment. For all those reasons we don't think it's the right solution."
Mr Drakeford said the UK Government's own advice to people in English hotspots was not to travel in or out of them and that he was "very happy to reinforce that message".