Japan's Nikkei 225 index fell 2.7%, South Korea's Kospi shed 1.8% and Hong Kong's Hang Seng dropped 2.4%.
On Sunday, many French and Greek voters backed politicians opposed to current plans to cut state spending.
The worry now is that the eurozone's hard-fought fiscal compact, aimed a cutting debt levels, may fall apart.
"Investors are concerned on whether the eurozone fiscal
compact will survive," Arjuna Mahanedran of HSBC Private Bank told the
BBC. "That is key to solving the region's debt crisis."
On Sunday night, France elected a new President, the
socialist Francois Hollande, who has said he wants Europe's economies to
cut debt levels by focusing more on growth rather than austerity.
Germany v France?
France and Germany have been at the heart of attempts to solve
the eurozone debt crisis, and Mr Hollande's predecessor Nicolas Sarkozy
worked very closely with German Chancellor Angeal Merkel on bailout
plans and austerity measures.
They were the driving force behind the eurozone's fiscal compact.
However the election of Mr Hollande now brings that partnership into question, analysts said.
Mr Hollande has said that he will re-negotiate the pact that
enforces greater financial discipline among eurozone economies, and has
previously said that it was "not for Germany to decide the future of
Europe".
Mrs Merkel by contrast has maintained that there will no renegotiation of the pact.
Analysts said that given the different approaches it may
become difficult for eurozone members to agree on further measures
required to solve the crisis.
"The Merkel-Hollande initiative will never materialise due to
Hollande and Merkel being polar opposites with no chance to agree on
anything," Jeff Sica, president of SICA Wealth Management.
'Ticking time bomb'
In Greece, the country's governing parties, Pasok and New Democracy, lost their parliamentary majority after Sunday's election.
The
left-wing coalition Syriza, which has been against the government's
austerity measures, emerged as the second-biggest party with more than
16% of the votes.
Analysts said the Greek result showed that voters were deeply
unhappy with the emergency bail-out plans agreed between Greece, the
European Union and the International Monetary Fund.
They said the new government may try and renegotiate some of
the austerity measures, which include government spending cuts, state
job losses and tax increases, in a bid to appease voters.
"The Greek side of things is a ticking time bomb," Justin Harper of IG Markets told the BBC.
"The fear now is that all the hard work that had gone into
getting the second bailout package may start to unravel and we can
expect some real pressure on stock markets in the coming days."
Risk aversion
The declines in Asian markets were not just limited to stocks.
The euro fell to a three-month low of $1.2955 against the US dollar in Asian trade.
The price of oil also declined. Brent crude was down by 1% to
$111.95 per barrel and light crude fell by 1.7% to $96.80 per barrel.
Analysts said investor sentiment was also dented by
weaker-than-expected jobs data from the US on Friday, which showed that
jobs growth in the world's biggest economy slowed during April.
"When you take the unknown of what is happening in Europe and
combine it with the slowdown in the US jobs growth, that is pushing
people away from riskier assets," said David Lennox of Fat Prophets.
"At the moment, given the political uncertainty in Europe, you have to count the euro as a risky asset," he added.