There is undoubtedly niShedha (ban) on traveling across the ocean. This is clearly stated by Parashara, who has detailed the code of conduct for Kali Yuga and by others like Yagnavalkya. Bodhayana lists samudrayAna in kalivarjya dharma prakaraNa (samudrayAnaswIkAraH kamaNDaluvidhAraNam). There however are, various instances of samudrayAna and dwIpAntaragamana obtained in our scriptures: like in Mahabharata where Nakula, the fourth pANDava undertook samudrayAna [tatassAgarakukShisthAn mlecChAn paramadAruNAn etc - sabhAparva.]
Going by the pramAna:
dharmaM jignAsamAnAnAM pramANam paramaM shrutiH |
dwitIyaM dharmashAstraM tu tR^itIyaM lokasaMgrahaH ||
The greatest of pramANa for anything related to Dharma is Sruti. Next is Dharma Shastra (Smrti) and Sampradaya follows next. Though there seem to be no direct ban on samudrayAna by the Sruti, due to pramANas in the other two, samudrayAna can be considered as a banned activity.
Except for the purpose of Setu Yatra, if one travels continuously for more than three days on sea, such an activity becomes samudrayAna and one indulging in it accrues sin:
mAsaM kArAgR^ihe vAsaH naubhiH yAtrA dinatrayam |
mlecChavAse tathA pakshaM yo vaset sat u pAtakI ||
But even this is not accepted by all as the reference here is only to travel in a boat or ship and there is no mention of sea/ocean anywhere in this smrti vAkya. Due to lack of pramANa in favor of samudrayAna under any circumstance, some scholars reject the above interpretation. Mleccha samparka is another reason for the ban on samudrayAna.
prAyashchitta for this act is prescribed by Bodhayana Sutra -
traivArshika prAyashcitta (samudrayAnaM brAhmaNaswApaharaNam etc.)
Also, punaH saMskAra is recommended by bodhyAyana:
saurAShTrasindusauvIRAnavantim dakShiNApatham |
etAni brAhamNo gatvA punaH samskAramarhati ||
Here, `punaH samskAra' should be understood as abda prAyashchitta [trimshatkR^icChA abdasa~NjnakAH], detailed in sarvaprAyashchitta prakaraNa of various Smrtis:
sarvajanmArjitAnIha bhrUNahatyAdikAnapi |
sarvapApAni nashyanti kR^iCchrairdwAdashavArshikaiH ||
There are various pratyAmnAya-s prescribed for kR^iCchra, important being godAna. Others include: reciting a thousand Gayatri-s, performing 200 prANAyama-s, feeding twelve learned brAhamaNas, twelve ritualistic baths in puNya tIrtha-s, sahasrahuti tila homa, reciting swashAkha of the Veda etc. SamudrayAna falls under the category of an upapAtaka.
Quoting varAha purANa:
mahApApAdi pApAni gItAbhyAsaM karoti chet |
kvachit sparshaM na kurvanti naLinIdalamambhasA ||
gItArthashravaNAsakto mahApApayuto.api vA |
vaikuNThaM samavApnoti viShNunA saha modate ||
gItARthaM dhyAyate nityaM kR^itvA karmANi bhUrishaH |
jIvnamuktaH sa vij~neyo dehAnte paramaM padam ||
One who does anusandhAna of gItA shAstra is freed of all sins. Hence, study of gItA was recommended in addition to prAyashchitta by H H Sri Sri Sachcidananda Shivabhinava Nrsimha Bharati Mahaswamigal of Dakshinamnaya Sringeri Sharada Peetham to the then Maharaja of Mysore who undertook Samudrayana to reach England and thereby acquired mlecCha samparka.
Mahamahopadhyaya Tarka Vachaspati Sri Taranatha concludes that due to the lack of any direct prAyashcitta for samudrayAna itself and the need for naukAyAna for prescribed activities such as setuyAtrA, inference is that it is not merely samudrayAna which is an upapAtaka but the resulting mlecCha samparka which grants it the status of upapAtaka. Thus, he negates the need for prAyaschitta if samudrayAna does not result in mleccha samparka and if the brAhamaNa does not give up his nitya karmas during the samudrayAna.