Abdul Kader Mullah is be hanged for crimes against humanity during the 1971 war of independence with Pakistan.
His party Jamaat-e-Islami called the two-day nationwide strike, describing the trial as politically motivated.
However, his sentence was welcomed by his opponents.
Schools and businesses were shut on Wednesday. Motorways were empty and bus services were also suspended, reports said.
Police fired rubber bullets and tear gas to disperse protesters in the western city of Rajshahi, while the protesters threw bricks.
One man was killed in southern Bangladesh after being hit by a stone thrown by protesters, officials said.
Border guards have been deployed in the capital to manage law and order, the BBC's Anbarasan Ethirajan in Dhaka reports.
Dhaka's busy commercial districts are mostly deserted, our correspondent adds.
Mullah was sentenced to life imprisonment for war crimes by a special tribunal in February.
Human rights groups have said the tribunal falls short of international standards.
The sentence at the time prompted angry protests from critics who said it was too lenient. His supporters, meanwhile, said the charges were politically motivated
Mullah appealed to Bangladesh's Supreme Court against his life sentence, while the state appealed for a tougher sentence.
Clashes broke out on Tuesday after the court sentenced him to death.
Many buses and cars were vandalised in the southern region of Chittagong, our correspondent says.
Clashes have broken out across Bangladesh during a strike called by the largest Islamist party, a day after one of its leaders was sentenced to death for war crimes.