Latex sensitivity is a serious problem. It is estimated that up to 3.6 million people in Britain are already sensitised, including 17% of nurses. It costs employers, such as the NHS, millions of pounds in days lost through sickness. You should try and speak to whoever is responsible for health and safety issues in your organisation.
Once you have told your employer that you have a health issue, it is their responsibility to consider what can be done to safeguard your welfare. They also have a duty to keep information about your health confidential (unless you give them permission to disclose it) so not everyone has to be told of your sensitivity to latex, if that's not what you want. Your employer simply has to take reasonable steps to prevent you suffering injury – including in this case making sure no balloons are used.