Recent polls show almost 60% will say "no" to the document on Wednesday.
After France's rejection of the charter on Sunday, a Dutch "no" would leave Europe's leaders with no clear backup plan for what to do when two of the 25-nation union's members say they will not approve the new ground rules.
Voting booths were to open at 7.30am local time and close at 9pm, with first results projected half an hour later.
"Let's not let ourselves be led by polls," said Prime Minister Jan Peter Balkenende on the television programme Netwerk, in a last-minute plea for a "yes" vote. "Let's hope that when they're in the voting booth, people think about all the people that say this constitution would be a positive development."
Dutch supporters - including both Balkenende's conservative government and the main opposition Labour Party - say the constitution would streamline decision making in the union, and create a single foreign minister to give Europe more sway in international affairs.
Opponents fear that the Netherlands, a nation of 16 million, will be engulfed by a super state with headquarters in Brussels and dominated by Germany, France and Britain.
That could mean the end of liberal Dutch policies such as tolerating marijuana use, prostitution and euthanasia.
Still, other voters will say "no" to voice their discontent with the unpopular Dutch government.
[A no vote] "will be a success for democracy, but a drama for the constitution"
Editor of Dutch magazine Quote
Many are angry about price increases that followed the introduction of the euro in 2002, and some fear that Turkey will soon be admitted to the union, worsening tensions between Dutch Muslims and the non-Muslim majority.
Jorg Kelder, editor of the Dutch magazine Quote, said politicians were reaping what they had sown.
"People see that they were bamboozled" by the price increases after the introduction of the euro, he said. A "no" vote "will be a success for democracy, but a drama for the constitution," he said on television programme Nova late on Tuesday.
Following the French
Balkenende repeated that a Dutch "no" would not mean that politicians would resign.
In France, Prime Minister Jean-Pierre Raffarin resigned and was replaced by interior minister Dominique de Villepin in the aftermath of the rejection.
Some analysts expect Dutch "no" voters to be emboldened by the outcome in France, since the Dutch will avoid the stigma of casting a lone veto.
Balkenende (R) is urging voters
to say "yes" in the referendum
Peter Kanne, of the TNS Nipo polling agency, said the French "won't be a decisive factor" on the turnout.
"Mainly, people will do what they intended to do anyway," he said in a telephone interview. TNS predicts 59% of Dutch will vote "no", versus 55% in France, with a turnout of below 50%.
Aurore Wanlin, an analyst with the London-based Centre for European Reform, said the EU leaders' meeting in Brussels in mid-June was unlikely to pronounce the treaty dead.
But "when two member founding states vote no, it looks like a big crisis", she said.
So far, nine countries have ratified the constitution - Austria, Hungary, Italy, Germany, Greece, Lithuania, Slovakia, and Slovenia by votes of their parliaments; and Spain by a referendum.
Efforts to ratify the treaty will continue, in the hope that the French and Dutch remain the only members to shun the treaty by the time the process ends in October 2006.