* Dollar downtrodden after suffering worst year since 2003
* Asian shares consolidate past gains as Japan takes holidays
* Commodities benefit from dollar weakness, China demand
SYDNEY, Jan 2 (Reuters) - The euro stood within striking distance of its 2017 peak on an ailing U.S. dollar on Tuesday, while Asian stocks began the new year close to their highest in a decade.
Sentiment was helped by news that North Korea had offered an olive branch to South Korea, with Kim Jong Un saying he was "open to dialogue" with Seoul.
Yet activity was sparse, with Japan on holiday and many investors on an extended break. MSCI's broadest index of Asia-Pacific shares outside Japan was a fraction firmer after rising by one-third in value last year to heights last visited in 2007.
Japan's Nikkei also had a bright 2017 with gains of 19 percent.
While Wall Street had ended Friday with modest losses, it was still a bumper year for U.S. stocks.
The benchmark S&P 500 climbed 19.5 percent during 2017, while the Dow added 25.2 percent and the Nasdaq 28.2 percent, all the best yearly performances since 2013.
Still to come on Tuesday was the Caixin survey of Chinese manufacturing which is expected to show a slight slowdown as a punishing crackdown on air pollution and a cooling property market weigh on the world's second-largest economy.
The official Purchasing Managers' Index (PMI) released on Sunday dipped to 51.6 in December, from 51.8 in November, though the index of non-manufacturing rose to a three-month high of 55 from 54.8 in November.
In currency markets, the dollar remained out of favour having hit a three-month low against a basket of its peers on Friday. That brought its losses for 2017 to 9.8 percent, its worse performance since 2003.
Its pain was the euro's gain, with the single currency enjoying its strongest year against the dollar in 14 years. Early Tuesday, the euro was firm at $1.2013 and just off a three-month top of $1.2028.
Bulls were now eyeing the September peak of $1.2092, a break of which would take the euro to ground last trod in late 2014.
The euro had already broken major resistance on the yen to reach highs not seen since late 2015 at 135.51, leaving the dollar struggling at 112.74 yen.
A major hurdle for the dollar will be Wednesday's release of minutes from the Federal Reserve's December meeting when it raised interest rates. Two policymakers voted against the move amid doubts inflation would accelerate as hoped.
"With the market pricing in a 68 percent chance of a March hike and two hikes for 2018, there will close inspection to assess just how shaky their confidence is for any pick-up in inflationary trends," said Chris Weston, chief markets strategist at broker IG in Sydney.
"That said, the U.S. dollar is underloved and oversold and it won't take much to promote a bout of profit-taking from the shorts."
The skid in the dollar, combined with strength in Chinese demand, has benefited commodities priced in the currency.
Copper stood tall at $7,251.50 a tonne, having risen 31 percent in 2017 to a four-year top, while aluminium amassed gains of 34 percent.
Gold was 0.2 percent firmer at $1,305.62 an ounce, after advancing by 13 percent in 2017 for its best performance in seven years.
Brent crude oil futures ended the year with a 17 percent rise, while U.S. crude rose 12 percent on strong demand and declining global inventories.
Early Tuesday, Brent was steady at $66.62 a barrel while U.S. crude eased 17 cents to $60.25.
(Editing by Richard Borsuk)