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.

The US Commerce Department said the economy grew at an annualised pace of 1.4% in the January-to-March period.
The rate was an upward revision from the previous estimate of 1.2%, which itself was an increase from the original reading of 0.7%.
However, it still marks a slowdown from the final quarter of 2016, when the economy grew at a rate of 2.1%.
The latest growth figure was helped by an increased estimate for growth in consumer spending, which was revised up to a rate of 1.1% from 0.6%.
"The economy is expanding at a solid, if unspectacular pace," said Gus Faucher, chief economist at PNC Financial Services.
Growth estimates in the first quarter are often weak, a quirk some say is due to the difficulty of measuring the effect of seasonal changes.
Thursday's update bolsters the perspective of the Federal Reserve, which increased interest rates in June.
Policymakers at the time said they did not believe the slowdown in the first quarter was the start of a trend, pointing to one-off factors, including a relatively mild winter.
Stronger-than-expected trade figures published Wednesday also led some to predict better growth in the second quarter.
Even so, many say growth for the year is all but certain to fall short of the 3% goal outlined by US President Donald Trump. Mr Faucher forecasts growth around 2.2% for the year.
The International Monetary Fund this week cut its forecast for US economic growth, in part citing uncertainty over the chances for tax reform and infrastructure spending, policies that many say could provide an economic boost.
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.
[NEXT_CONCEPT]
The US economy grew at a faster pace than previously thought in the first three months of the year.