New Finance Minister Mthuli Ncube sent an appeal on Twitter, sharing a mobile payment account number.
An emergency has been declared and public gatherings banned in Harare to prevent the spread of infection.
In 2008, a cholera outbreak killed some 4,000 people and at least 100,000 people fell ill.
This was a key factor in persuading President Robert Mugabe to agree a power-sharing government with the opposition, as the government did not have the money to deal with the outbreak. The current outbreak began on 6 September after water wells were contaminated with sewage in Harare.
Tests found the presence of cholera and typhoid-causing bacteria which has so far infected over 3,000 people, Health Minister Obadiah Moyo told reporters on Thursday. According to the World Health Organization (WHO) situation report, patients were not responding to first-line antibiotics.
"Relevant medicines should be purchased as a matter of urgency as soon as resistance patterns have been ascertained," it said.
WHO also said the disease has spread to five of the country's 10 provinces. The cholera outbreak can be traced to Harare city council's struggle to supply water to some suburbs for more than a decade, forcing residents to rely on water from open wells and community boreholes, according to Reuters news agency.
Health officials are advising people to wash their hands regularly, drink only safe water, wash food, cook it thoroughly and avoid shaking hands.
The government-controlled Herald reports that the crowdfunding campaign has already received some backers.
They include telecommunication giant Econet Wireless, which has contributed $10m (£7m) and the Zimbabwe Red Cross Society $250,000 (£190,000).
However, some Zimbabweans have taken to social media to condemn Mr Ncube's plan.