The storm was a category two hurricane, the second level on the five-step Saffir-Simpson scale of intensity.
Russell Stockton, who operates Dolphin Docks, a dolphin-viewing tour company, on South Padre Island said his dock had been been torn down.
"It's about $50,000 worth of damage so far," he said.
The storm steered clear of most offshore drilling rigs and production platforms in the Gulf of Mexico.
US crude oil prices, affected earlier in the week by worries about possible storm damage, hit 6-week lows on Tuesday and fell further on Wednesday to below $126 a barrel.
A hurricane warning was issued for the southern Texas coast as far north as Corpus Christi.
Category three to five hurricane are considered to be the most dangerous, but a category two hurricane is still capable of causing damage to poorly built or fragile dwellings such as mobile homes.
The National Hurricane Centre (NHC) said Dolly could dump up to 51cm of rain on low-lying areas in South Texas and northeastern Mexico in the coming days.
The prospect of heavy rains and a storm surge of sea water pushing back upstream has spurred concerns that levees holding back the Rio Grande river could be breached, causing widespread flooding.
Rick Perry, the Governor of Texas, put 1,200 National Guard troops on alert and issued a disaster declaration for 14 low-lying counties.
State officials said that they would not order mandatory evacuations unless Dolly reached Category 3, with wind speeds of over 178kph. Hurricanes usually weaken rapidly over land.
About 250 buses stood by in the inland city of San Antonio to evacuate coastal residents if needed.
In Cameron County near the Mexico border, officials expected more than 50cm of rain. "That's going to do a number on our county," Johnny Cavazos, the county's emergency management co-ordinator, said.
|Residents in south Texas boarded up businesses before the storm hit [AFP]
The levees holding back the Rio Grande river held under similar conditions during Hurricane Beulah in 1967, but have "seriously deteriorated" since then, he said.
Texas State Police Captain Joe Gonzalez, who heads the combined emergency management system in Brownsville, said he was confident the levees would hold.
More than 27,000 customers were without electricity in South Texas, most of them in Cameron County, according to the power company.
Officials late on Tuesday closed a causeway connecting Brownsville with South Padre Island. Many residents of the island had already left.
In the Mexican city of Matamoros, across the border from Brownsville, streets began to flood with strong gusts of wind.
Four Mexican fisherman were missing off the Yucatan Peninsula after Dolly swept through the area on Monday, port authorities said.
The 2008 Atlantic hurricane season is already a month ahead of schedule. On average, the fourth tropical storm of the six-month season does not occur until August 29.
Dolly, this year's fourth, formed on July 20.