http://www.melbourne.vic.gov.au/community/health-support-services/social-support/Pages/begging.aspx
Begging is illegal under the Summary Offences Act and enforcement is the responsibility of the police.
The recent mayor election seems to have ramped up the enforcement against the crime of begging recently in Melbourne. With fines being applied to those who beg in the streets (according to one of them, it is about ~$230 per offence)
While I imagine there is some who are professional beggars, this could potentially run the risk of placing a criminal conviction to those who genuinely needs help. My concern would be the difficulties of getting a job after getting a conviction or jail-time for the crime of begging. Plus the economic cost of jailing such offenders.
Media
Other Parties stances on this
On a side note, there is this book http://www.melbourne.vic.gov.au/sitecollectiondocuments/helping-out-booklet-map.pdf if you need some information on resources available in Melbourne if you are in need of help.